INFINITE GROUPS G. A. Noskov, V. N. Remeslennlkov, and V. A. Roman'kov

UDC 512.543+512.544

Abstract group theory as an independent area of mathematics has been around for a little more than 50 years. At the present time it can be characterized as an intensively developing and, at the same time, mature area of scientific knowledge, proving to be an ever increasing influence on the development of mathematics and of natural sciences as a whole. When speaking of the modern state of group theory it would be impossible not to mention the outstanding mathematician M. I. Kargapolov (1928-1976)o To him are due a number of profound results in various directions in group theory. Many important researches by Soviet specialists on group theory in the last 20 years were brought to llfe by the problem, the support, the inspiring attention, and the energetic participation of Kargapolov. Our purpose is to present the development of the theory of infinite groups in the seventies and to give the state-of-the-art. The survey has been written mainly from the material in the papers reviewed in Referativnyi Zhurnal '~atematika" during 1971-1977, In the selection of the material we were guided not only by our concept of its importance and our tastes, but also by the presence of sufficiently complete survey articles and monographs on certain sections of the theory of infinite groups. It is precisely for this reason that we did not here cover such areas as linear, Abelian, ordered, topological groups; lattices, radicals, group presentations, group rings, etc. Several international and all-union conferences on group theory were held during the period being surveyed. Twice, in 1973 and in 1976, there were published the Kourov Notebooks, viz., collections of unsolved problems in group theory. There appeared a number of monographs and textbooks, in particular: M. I. Kargapolov and Yu. I. Merzlyakov, Foundations of Group Theory [in Russian], Pts. i, 2, Novosibirsk State Univ., Novosibirsk (1968, 1969), pp. 1-196~ 199-365. The authors thank everyone who helped them in the writing of this survey. 1.

Free Constructions

Investigations of groups prescribed by generators and defining relations constitute an independent extensive area traditionally named combinatorial group theory. The theory is characterized by a specific circle of problems, many of which have their analogies in algebraic topology. The method mentioned for prescribing groups also originates from topology, e.g., the fundamental groups of topological spaces are prescribed in that way. I= occurs in the classical works of Poincar~, Dehn, and Nielsen, and was first explicitly defined by Dyck in 1882-1883. By its very name the theory is necessarily a combinatorial method of investigation, consisting in an analysis of formal words in a specified alphabet, in canonic forms of writing them, in operating with the lengths of these words, etc. Recently the theory was essentially enriched by new approaches to the solving of problems. Among them is the geometric method, revived by Lyndon and his students~ due to Poincar~ and Dehn; a part of the monograph [697] is devoted to it. The efforts of Bass and Serre conferred great value to the language of group action on a graph (see [203p 308]). We also note Fox's free derivation (see [119]), cohomological methods (see Sec. 9), etc. The foundations of combinatorial group theory were set forth in the textbook by Magnus, Karrass, and Solitar [138] and in the monograph by Lyndon and Schupp [697~. Also see the lectures of Cohen [431] and Johnson [620].

157,

Translated from !togi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 17, pp. 651979.

0090-4104/82/1805-0669507.50

9 1982 Plenum Publishing Corporation

669

Free Constructions. The basic constructions studied in combinatorial group theory are free product, free product with amalgamation and its generalization, viz., tree product, and also the so-called HNN-extension. We shall define them and, henceforth, for brevity, we shall call them free constructions. If A~, i~! , are groups with nonintersecting generator sets, then their free product G=i~ ~ Ai

is the group prescribed by the union of all generating and defining relations of

the factors. Let T be a connected graph without cycles (a tree) whose vertices are the groups At, i~l , and to each edge e~E

of it, joining vertices Ae~11, Ae(21, let there be associated a group He

embedded in these vertices by the monomorphisms ~e(1)'~e(2) , respectively.

A group G with the

following presentation:

G = < 161 * A "l, ~e(1)He~e(2)He, is called the tree product of groups

erE )

A~, igl, with amalgamations I/e, e6E 9 It is well known that

the subgroups Ai. iEl, are embeddedin G relative to natural maps; moreover, Aeo) AAe(2)=ffe for all

erE.

In the special case when T is a tree with two vertices we obtain a free pro-

duct with amalgamation O = A I ~ A ~ . Suppose that pairwise isomorphic subgroups Li~iLi, iEI , have been picked out in a group G.

The group G* with presengatlon O*-----
is called a HNN-extension with base G and associated subgroups L~, ~iLi .

The inventors of

this construction, Higman, B. Neumann, and H. Neumann (HNN), proved that group G is naturally embedded in group G*. Free constructions admit of a clear topological interpretation.

Let Xx, Xa be linearly

connected topological spaces with a nonempty linearly connected intersection Xo. we denote the fundamental group of space X i, i = 0, i, 2. morphisms Go-+Gi, i= I, 2 , are embeddedings. is represented as G=G~G% G2.

By G i

We assume that the natural homo-

Then the fundamental group G of space

X=XIUX2

The assertion made is a special case of the Zeifert--van Kempen

theorem (see [142]) from which there is extracted an interpretation for HNN-extensions.

Let

Xx, Xa be open homeomorphic linearly connected subspaces of a linearly connected space X, whose fundamental groups Gx, G2 are isomorphically embedded in the fundamental group G of space X. Attaching the space XIX[0, I] as a "handle" to X, i.e., identifying XIX{0}

with Xx and XIX{I}

with X2, we obtain a space X* whose fundamental group G* is an HNN-extension with base G and associated subgroups Gx and G2. The spelling rules for canonic words of the representative elements of free constructions are well known.

An element of a free product with amalgamation

as g=huoblalb~...Un_ibn , where

G=A*BH

is uniquely written

hg/-/, a~, b~ are representatives of the cosets of groups A, B,

respectively, with respect to H; moreover, as the representative of the trivial coset we select 1 and ai' bi ~ 1 for i ~ 0, n.

An element of the HNN-extension O * = ( G, ti t-ILt=~L >

8n is uniquely representable as g----gote'g~e'g~...r g, , where

g0~O; if e~-~---I , then gi is the

representative of the coset with respect to subgroup L in G; if r sentative of the coset with respect to subgroup are no subwords of form tel~-e 9

670

, then gi is the repre-

~L in G, and in the notation used here there

In order to stress once more the connection of combinatorial group theory with topology, we present a translation into algebraic language of Poincar~'s famous conjecture: Every compact connected simply-connected 3-manifold is homeomorphlc to the 3-sphere.

This conjec-

ture is the stumbling block on the path to the classification of compact 3-manifolds (the case of 2-manifolds has been completely exhausted (see [142]).

It is well known that a compact

connected orientable surface of genus m has a fundamental group ~m ~ m = ( at, bl..... am, be;

~[al,bl]=l)

.

with the presentation

Stallings (1962) and Jaco (1962) established that the

i=l

assertion of the Poincar~ Conjecture is true in this and only this case if the epimorphism ~:~m-+FmXFm

, where F is a free group of rank m, can be factored into a free product, i.e., m can be represented as a superposition of the eplmorphisms $:~-~A*B, A, B~l, x:A*B-+F~XF m. Structure Theorems. By a well-known theorem of Kurosh an arbitrary subgroup of a free product of groups is as well a free product of factors of specific form. The analogous assertion for the subgroups of a free product with amalgamation of groups is invalid. The first step toward describing them was taken by H. Neumann by defining the concept of a generalized free product~ However, this concept did not allow a satisfactory investigation of the properties of such subgroups. Therefore, a search was undertaken for new approaches which would give good results. Karrass and Solitar [631] proved the following fundamental structure theor~. an arbitrary subgroup of a free product with amalgamation

G=A*uB

9

Let H be

Then there exist systems

of representatives {~}, {s~} of group G with respect to the binary modules (H, A)~ (H, B), respectively, and a set of elements rt, re, ..~ such tha~: I) H is generated by the elements

H=:,=H.~t~Atj1, H~=flns~Bs~'; 2) the subgroups with amalgamations U~-~-Hng~Ug~z U~-Hn$:~Us~I; 3)

r,, re, ... and by all possible subgroups

H~,H~

generate in H a tree product T

is some HNN-extension with base T.

H

A generalization of this theorem to tree products is given

by Fischer [485]. In their next paper [632] Karrass and Solitar obtained an analogous theorem for subgroups of HNN-extensions.

An arbitrary subgroup H in a HNN-extension G* with base G is itself a HNN-

extension whose base is a tree product of subgroups of form

gGg-~flH ,

while the associated sub-

groups lie at the vertices of the base and either coincide with them or have the form gLig-1flH, where the L i are the associated subgroups of group G e. Schupp's paper [869] also was devoted to subgroups of HNN-extensions. Strengthening of the Karass--Solitar theorems are contained in their Joint paper [637] with Pietrowskl. As we see from the theorems cited, it is convenient to examine free constructions Jointly, Under certain approaches they are special cases of broader concepts. The theory of groups acting on trees, essentially developed by Bass and Serre, yielded new effective methods of investigation for combinatorial group theory. They have been presented by Serre [203] and others [308, 878].

As Serre showed, free groups are precisely those groups Gwhich admit of an action on some tree, under which each element

ggG, g ~ l

, does not stabilize even one vertex of the tree.

Hence, in particular, it follows automatically that an arbitrary subgroup of a free group is free as well. A graph of groups (@, F) is a connected graph F and each edge

in which: a) with each vertex

v~(F)

egE(F) there are associated groups Gv and G e, respectively; b) for any e~E(F)

there are indicated embeddings of group G e in groups Gv and % ~

where the vertices v and w

671

are joined by edge e (if v = w, then the embeddings are, in general, different). introduced the concept of the fundamental group relative to the maximum tree T of graph T,

For each edge e from

r\T

r.

~(~, F, T) of the graphs of groups (~, ri

By G(T) we denote the tree product relative to

the group G(T) contains a pair of subgroups isomorphic to G e,

so that we can construct a HNN-extenslon. we obtain the group

Serfs

After all such HNN-extensions have been produced,

~(~, F, T).

A fundamental structure theorem, proved by Serre in [203], states that the representation of group G as

~(@, r, T) is equivalent to the definition of the action of G on a tree X (a

"universal covering

(~, F, T)").

H is a subgroup of O ~ ( ~ ,

r, T}

Hence, as a corollary we obtain a theorem on subgroups: If , then H too acts on X; therefore,

H=~(~', F', T') for the

factor group (~',r') . Stalllngs (1968) introduced the concept of a bipolar structure on a group (partitionings into nonintersecting sets with specific properties) and proved that a group possesses it if and only if it is representable as a nontrivial free construction. As corollaries of this theorem there follows a number of assertions on the structure of subgroups of free constructions. Topological methods and the formalism of the theory of groupoids were used by Crowell and Smythe [446] to generalize the Karrass~olltar theorems. A topological approach to the description of subgroups of free constructions is contained as well in Chipman [419]. Ordman [775, 776] derives a general theorem constructions, from which follow, in particular, theorems of Kurosh and Grushko are proved by new of progressiveness of Petresco) and by Chiswell

on the structure of the subgroups of free the theorems of Kurosh and Grushko. The methods by Boydron [366] (using the concept [424] (on the basis of the Bass--Serre theorem).

Characterization of Free Constructions. In [878] Serre considered groups with property (FA): every tree on which G acts has a fixed point. He proved that a countable group G has property (FA) if and only if: a) G does not have an infinite cycllc factor group; b) G is not represented in the form G = AH*B, H # A, B; c) G is finitely generated. If the condition of being countable is not imposed on group G, then the statement made is true as well, except that condition c) must be replaced by the condition c') G is not the union of an ascending sequence Ox
Let K be an ordered Abelian group.

The length function on group G with values in a group K is a map following conditions:

(ixl+lyl_lxy-~l)~o implies x = y.

672

a) Ixl~0

and Ix[=0

; d) d(x,y)~d(x,z)

[ [: G-~K

responding to the 1 if and only if x - l; b) Ix-If=Ix[ ; c)d(x, y ) = ~

implies

d(y,z)=d(x,z) ; e) d(x,y)+d(x-~,Y-')~lxl=lYl

It is evident that these axioms aresatisfied by the usual lengthfunction

in a free group. Lyndon proved that under the additional condition IxI
On the subject of group

Various representations of a group as a free product with amalgamation are examined in

[777]. Properties of Subgroups of Free Construction. The following assertions and their generalizations were examined for free constructions: a) a finitely-generated normal subgroup has a finite index; b) a finitely-generated subgroup is finitely presented; c) the intersection of finitely-generated subgroups isfinitely generated. These questions were studied by Karrass, Solitar, Cohen, Goryushkin, Burns, and others (see the literature cited). Cohen [428] proved that an almost free group is a HNN-extension of a tree product of finite groups, in which the associated subgroups lie at one vertex. A number of statements (mainly, well known) on subgroups of free constructions were proved by topological methods by Tretkoff [923]. The Frattini subgroups of free constructions were studied by Tang and Allenby (see the literature cited). _Groups with One Definin~ Relation. The main stimulus for the investigation of onerelator groups is the fact that the fundamental groups of 2-varieties fall into =hls class~ The majority of the most important assertions on such groups have been known for a long time (see [138, 697]). The methods for investigating one-relator groups are based on the fact that they, as a rule, are represented as free constructions. A number of problems on one-relator groups were formulated by Baums!ag [324]. problems 5 and 9 have been solved in [486].

Of these~

The structure of one-relator groups with torsion was investigated by Fischer~ Karrass~ and Solitar [487]. Weinbaum [957], using geometric arguments, proved that a proper subword of the defining word of a (cyclically reduced) one-relator group does not define 1 in this group, Subgroups with identity in a one-relator group have been described by Chebotar' [230]. Other papers relevant to the present paragraph also exist (see the literature cited). Other Ques=ions. Higman attempted to make a nonaxiomatic characterization of torsionfree groups. He postulated that any torsion-free group can be obtained from infinite cyclic groups by means of forming free constructions and taking subgroups and amalgamations of ascending sequences of groups. Baumslag, Karrass, and Solitar [330] showed that there exists a continuum of finite-generated torsion-free groups not obtained by the method indicated. Nevertheless the following question remains open: Question i.

How can the class of torsion-free groups be described nonaxiomatically?

The class K of groups, obtained from free groups by forming free constructions with the condition that the amalgamated and the associated subgroups are free every time~ was studied by Cossey and Smythe [445]. We note that all one-relator torsion-free groups and a!l knot groups fall into class K.

673

r

Lewin [678] proved that the group G = Fn*FkF m is free if and only if the minimum number of its generators satisfies the equality r = n + m -- k.

The bases for the commutants of free products of Abelian groups are computed in [284, 490, 936-938]. 2.

Defining Relations Construction of Examples.

Let V n, n~2, be a variety of universal algebras whose signa-

ture consists of unary operations ub u~..... =n and an n-sty operation identities: a) (ax, a2..... an)%ui=ai

, b) (ast, aa~ ..... a a n ) % = a

.

~ connected by the

By An, r we denote the auto-

morphism group of a free albegra of rank r in varietyV n. Higman [581], using an idea of R. Thompson, proved that the group An, r (for odd n) or its subgroup of index 2 (for even n) is infinite, finitely presented, and simple.

Since these groups are pairwise nonisomorphic for dif-

ferent n, by the same token we have constructed an infinite series of groups with the properties mentioned. Remeslennikov gave the following simple example of a finitely-presented pro-p-group G whose center is not finitely generated.

Group G is prescribed by the generators a, b, xl, x~, x1',x~',

and the defining relations:

1) [X,,X~]rl, i,k=l,2; 2) a ~ = a

x l

, bx i = b ,

i=1,2;

3) a'=ab, [a, bl=lb,~l=ix,,~l=t, 2 4) a'-*.-*,+*,*,=a~ -=*~+.I= 1.

i=1,2;

By C we denote a subgroup of group G, generated by the elements c n = b x~x'-1), nfiZ

.

Then C is

a free Ableian pro-p-group of countable rank, lying in the center of group G. If in Remeslennikov's example we replace the word "pro-p-group" by the word "group," we obtain an example of an abstract flnltely-prasented group whose center contains a free Abelian group of countable rank, i.e., is not finitely generated. An earlier and more complex example of such kind, also due to Remeslannikov, is contained in his paper [179]. In his address at the Sixth All-Unlon Symposium on Group Theory at Cherkassy (Sept., 1978) Ol'shanskli announced the construction of the following examples: i) an infinite nonAbelian torsion-free group any proper subgroup of which is cyclic and, moreover, any two maximal subgroups of it have a trivial intersection; 2) an infinite group any proper subgroup of which has a prime order and, moreover, any two subgroups of llke orders are contained in each other. By the same token counterexamples are given to the well-known conjectures: i) O. Yu. Schmldt -- on the finiteness of a non-Abelian group all of whose proper subgroups are finite; 2) S. N. Chernlkov -- on the existence of an Abelian subgroup of finite index in a group with the minimal condition; 3) R. Beer -- on the almost polycycliclty of groups with the maximal condition. Ol'shanskii's results have been presented in his paper: "Infinite groups with cyclic subgroups," Dokl. Akad. Nauk SSSR, 245, No. 4, 785-787 (1979). The proofs are carried out in the language of diagrams (in the sense of [697]). Embedding Theorems. Higman's theorem on the embedding of an arbitrary recurslvelypresented group into a finitely-presented group has been proved by a new method by Aanderaa [266]. It has been strengthened in [42, 356]. A number of theorems on embedding in finitely-presented groups were proved by Baumslag, Cannonito, and Miller [329]. In particular, they established such embeddings for countable locally polycycllc-by-finite groups and for countable linear groups. By the Higman--Neumann--Neumann theorem (1949) an arbitrary countable group G is embeddabla in a 2-generator group H. The subsequent investigations in this direction can be classified as follows. 674

i) Embeddings with preservation of specific properties of group G. The main results here relate to the preservation of the properties of solvability, nilpotency (see Sac. 4 regarding this), and torsion. Hickin [572] proved that a countable ~-group generated by elements of bounded orders is embedded in a 3-generator ~ -group (under this embedding a group of bounded exponent is embedded into a group of bounded exponent). Phillips [793] strengthened Hickin's statement, proving the embeddability of a countable ~-group in a 2-generator ~-group. Also see [872]. 2) Embeddings with prescribed requirements on group H. The most interesting result is that of Goryushkin [58]: an arbitrary countable group G is embeddable in a 2-generator simple group H. Schupp [870] sharpens this by showing that as group H we can take a suitable factor group of the free product H=Zm*Zn, Im]>2, In]>2. Embeddings in Hopfian and non-Hopfian groups are dealt with in [745] and [450], respectively. 3) Description of SQ-groups. A group is called an SQ-group if any countable group can be embedded in a suitable factor-group of it. Papers [717, 754, 854] are devoted to seeking new examples of SQ-groups. The main results in this direction are given in Schupp's survey

[867], A number of interesting embedding theorems have been obtained by Hall [539]~ Small Cancellation Groups.

Let F be a free group with base X, ~-----{r~~,r2~I,...,r~~} be a

set of cyclically irreducible words from F, containing together with each of its elements all cyclic permutations of it.

A group G wi=h the presentation

Q----- has a reduction

measure % (is a i-group) if the length of the raducad subword in the product of any pair of elements r~tr~Ji j ( r ~ r j ~j) from R is strictly less than mitting of a presentation with reduction measure

%,m~n(iriI, Ir/I} o

%~1/5

As a rule a group ad-

is called a small cancellation group.

The study of such groups began with Dehn's work analyzing the solvability of algorithmic problems in them.

A number of interesting results in this area are due to Greendlinger.

In his survey [868] on small cancellation groups Schupp formulates the unanswered questions which in recent times have determined =he following directions of investigations in this theory. i) Equality of the elements and root extraction. Schupp [865], using the diagram method, gives a geometric proof of Greendlinger's theorem on the fact that an arbitrary work defining ! in a sixth-group contains a large par~ of a certain defining relation of this group. Greendlinger's theorem yields an algorithm solving the word problem in a sixth-groupo A stronger assertion -- solvability of the word problem in fifth-groups -- has been obtained by Lyndon (1966). In a group G wi=h reduction measure rhr=,..,rh>

l=I/8

relative to the presentation O=
let there be no words of form x t among ~he defining relations.

[499] showed, if the equality g divides s and g = xS/n.

n

= x

s

Then~ as Gowdy

is fulfilled in group G for some element g~G,

then n

Lipschutz [683] presents an algorithm resolving the question whether

a nontrivlal root can be extracted from an element of a slxth-group.

His result is made more

precise in [438]. 2) Conjugacy of elements and of their centralizers. Lipschutz [684~ 686, 687] has proved that different powers of an inflnlte-order element in a slxth-group are not conjugate, Similar results, as well as the cycllcity of the centralizer of the nonunlt element in a sixth-group, were obtained by Truffault [925-927]. The algorithmic solvability of the conJugacy problem in flfth-groups has been proved by Schupp (1968). Using geometric methods, Comerford [437] described real (i.e.j conjugate to their own inverse) elements of sixth-groups. Special results on this question are contained in the earlier works of Greendllnger, Gowdy, and others. Presentations of Groups. A finite description of a group in terms of generators and defining relations is called its presentation. A number of authors examined the connections

675

between different presentations of groups and Gruenberg [512] noted the following interesting questions in presentation theory. i) Let F/R,, F/R2 be different presentations of a finite group G. Will the presentation ranks of R~ and R~ coincide? (The minimum number of generators of R as a normal subgroup in F is called the presentation rank (or relation) rank of presentation F/R.) The answer is "no" in the class of all finitely presentable groups. Dunwoody [455] found a 2-generator presentation for the trefoil group G=, having a presentation rank 2. 2) Let r(G) be the minimum of the numbers of relations needed to define a finite group G. Is r(G) realized in a minimal presentation (i.e., one in which F and G have the same minimum number of generators)? These questions have an affirmative answer if the following statement is valid: 3) The difference between uhe number of generators and the presentation rank is independent of the presentation and, therefore, is an invariant for G. Let F/R be a presentation of a group G. A structure of a G-module is specified on the group R = R/R' in a natural manner; this module is called the rela._tlon module determined by the presentation F/R of group G. Under the assumption on module R the statement 3) is valid, i.e., the difference between the number of generators of presentation F/R of group G and the number of generators of module ~ is an invarlant for G if G is finite. Consequently, questions i) and 2) relative to R have affirmative answers if again group G is finite. This result is due to Gruenberg (1970) and is contained in his lectures [512] devoted specially to relation modules. We present the main results on relation modules of a finite group (see [512]). I) If F/R,, F/R~ are different presentations of group G, then RI, K2 are locally isomorphic as G-modules, i.e.,

R1|174

all primes p.

If the number of generators of

group F is not minimal (i.e., is greater than the number of generators of group G), then we even have R I ~ 2 2) Let mand.

9

R=A@P

It is not known if this is true for a minimal presentation. ,

where P is a ZG-projeetive module and A has no projective direct sum-

Then A is unique to within local isomorphism whatever relation module R is used and

whatever decomposition is taken. By Swan's theorem P |

We call A a relation core for G. m.

In particular, for a minimal presentation F/R the number

m is independent of the choice of R and is called the presentation rank m = pr(G) of G.

For

all solvable groups G, pr(G) = 0. 3) The minimum number of generators of R (the rank of R) can be calculated from a suitable finite image.

More precisely, there exists a prime p occurring in IOl

such that the

rank of R equals the rank of R/pK. The already-mentioned example of Dunwoody [455] shows that in the case of an infinite group the analogs of statements 1), 2), 3 ) a r e false for the relation modules.

Lyndon (1962)

proved that for any relation modules R,, Ra there exist free ZG-modules P,, P~ such that RI@pI~2eP2

.

In Dunwoody's example

module Ka is projective.

~ZG,~2@ZG~RI@ZG~ZG~ZG_ .

Hence, we get that

However, it is not free and cannot even be represented in the form

R2~ZG@/W for any module M. By the Auslander--Lyndon theorem (1955) a natural presentation of group G = F/R in the group A = AutR is exact.

Passi [784] strengthened this assertion, having proved that the an-

nihilator of module ~ in ring ZG is trivial.

Also see [i18, 748] on relation modules.

An open problem of Andrews and Curtis (1965): If O = < x b x2, ..-, x,;

rl,

r2..... rn>

is a

trivial group, then can a sequence of Nielsen transformations and conjugations of one of the

676

elements take a collection of relations into a collection of generators? This question was studied by Rapaport. The following problem of Waldhausen is related to the one mentioned. Question 2. Let O = F ~ I R b e an (n ~ l)-generator group~ Is it true that the normal subgroup R contains an element of some basis for group Fn? Other Questions. Birman [356] gave, in the language of ~he free derivation of Fox, a condition for the complementability of a given collection of elements up to a basis in a free group of finite rank. Krasnikov [118] used derivations with values in ring Z(F/R) to establish necessary and sufficient conditions of generation by a given collection of n elements of group Fn/R'. Dunwoody [454] proved that the Hopficity of group Fn/R' follows from the Hopficity of group G = Fn/R. Fox subgroups of free groups are studied in [527, 614]. are computed in [440, 522]. 3.

The centers of certain groups

Varieties

H. Neumann's book: Varieties of Groups [Springer-Verlag, Berlin--HeldelberE-New York (1967)] gave a summary of the previous development of this comparatively young area of group theory and, as noted, served as a compendium of problems for the future. The majority of them were solved before long, as was noted in a special survey by Kovacs and Newman [651], By A, An, Bm, Nk we denote the varieties of Abe!Jan, Abelian of period n, Burnside of period m, k-nilpotent groups, respectively. Bases of Identities. Ol'shanskii (1969) established the existence of varieties that are not finitely based, i.e., varieties whose identities do not follow from any one of its finite subsets. Infinite irreducible systems of identities were indicated by Adyan [2] and Vaughan-Lee [929]; moreover, the identities presented in [929] define an infinitely-based subvariety in (B4A N~) ~ Thus, it has been established that the cardinality of the set of all different varieties is a continuum. The simplest example of an inflnitely-based variety is B~B2; this was independently and simultaneously established by Kleiman [i12] and Bryant [393]~ It is of interest to note that infinitely-based varieties can take up a whole infinite interval in the lattice of varieties [393]. After the discovery of Infinitely-based varieties the problem of seeking varieties with finite bases of identities received an additional stimulus~ Investigations were carried out mainly in two directions: either the finite basability was established for all varieties corresponding to a given identity (hereditary finite basabiiity) or the question was clarified on the existence of a finite basis for identities true on a given group. The starting points here are the well-known theorems: Lyndon (1952) -- on the finite basability of nilpotent varieties; Cohen (1967) -- on the minimal condition for metabelian varieties; Oates--Powell (1964) -- on the existence of a finite basis for identities fulfilled on a finite group. Papers [243, 396, 731, 93!, 932] are devoted to generalizations of Cohen's theorem. Higman (1959) observed that any nilpoint variety N possesses a property stronger than finite basability: for an arbitrary finitely-based variety D the product ~D also possesses a finite basis of identities. In [383] this result is generalized to a subvariety from A2VNk In the general case the property of finite basability of factors does not carry over to the product; the first example of such kind was indicated by Vaughan-Lee [933], where the righthand factor in it was a variety A. Furthermore, as shown in [I12], for any nontrlvial finitely-based variety C we can find a positive integer k such that the product BkC is Infinltely-based. We note that the question on the preservation of finite basability under the union operation remains unanswered as yet; there are only partial results [390~ 392].* Bases of identities of concrete varieties have been described in [376, 377, 416, 497, 676, 892]. Generalizations of Cohen's theorem to varieties close to metabellan have been obtained in [94-96]. The Oates--Powell theorem is strengthened in [441]. Also see [372, 391~ 400,

768]. *Added in proof. A counterexample has been given in: Yu~ G~ K~lelman, "Identities and some algorithmic problems in groupsg" Dokl. Akad. Nauk SSSR, 244, No. 4~ 814-818 (1979).

677

Quasi-varieties of groups are dealt with in [41, 99, 166]. Budkin [41] proved that a system of quasl-identlties true in a free non-Abelian group does not have bases of a limited number of variables. Ol'shanskii [166] established that the quasi-identities of a finite group form a finite basis if and only if all Sylow subgroups of this group are Abelian. Solvable Varieties. Kargapolov and Churkin [103] proved that any solvable variety of groups, not containing A ~, is a subvarlety in BkNcBm for some k, c, m. In particular, finitely-generated torslon-free groups in such a variety are almost nilpotent. Groves [503] generalized this statement to a subvariety of products of a finite number of solvable and locally finite varieties. All this is the result of lengthy investigations to which papers [504-506, 520] relate within the period being surveyed. The first result in this direction was obtained by Shmel'kin (1968) who proved an analogous statement for subvarieties from NcA. It remains to add that it is invalid for varieties containing A 2, as the example ZwrZ shows. The lattice of metabelian varieties was studied in detail: first, its elements, indecomposable into a union of proper subvarleties, were described; second, the dlstributivity relations were investigated. The rightfulness of such an approach is substantiated by Cohen's theorem, according to which each metabellan variety is the union of a finite number of indecomposable varieties, and by the fact that nondistributivity violates the uniqueness of such a decomposition. Indecomposable metabellan varieties were studied in [37, 381, 397, 399, 400, 974, 975]. In the nilpotent case they have been exhaustively described by Bryce [399]. The nondistributivity of the lattice of nilpotent varieties was established by Higman; Roman'kov [192] gave an expllcit example for subvarieties from N~. The minimal bounds were achieved by Belov [36] by establishing the nondistributivity of the lattice of subvarieties in A2A N~, since the lattice of 3-nilpotent varieties, described previously in the works of Remeslennikov and Belov, is distributive. The distributlvity of the lattice of subvarieties from ApnAp was proved by Kovacs and Newman [650]. On the other hand, sufficiently narrow nondistributive lattices of nilpotent varieties are constructed in later papers [37, 382, 779], in particular, such as the lattices of subvarletles in A 2 A d A N ~ 9 Certain distributive lattices have been described in [37, 113]. Locally Finite Varieties. A variety generated by a finite group is called a Cross variety. Minimal non-Cross varieties have been named almost-Cross by Kovacs and Newman. They have proved that the following varieties are such:

A, A~, ApiN2ABq), p, q=#2 are dis~nct primes, Ap(N2AB4) , p=/=2, ApAqAr, p, q, r are di~mct gimes. Ol'shanskli [162] proved that the list presented exhausts all solvable almost-Cross varieties. This assertion was confirmed for special cases in [371, 444, 650]. Razmyslov [176] was the first to construct a series of unsolvable almost-Cross varieties were C belongs to the Kostrikin variety K of locally finite groups of prime period P P p and is a locally solvable but unsolvable variety. The construction is effected thus: In explicit form there is constructed a nonnilpotent variety of Lie algebras Lp_~,p over a field of characteristic p > 3 with a (p-2)-Engel identity, every subvariety of which is nilpotent. It is proved that on the algebras from ip_~,p we can introduce an operation by the Campbell-Hausdorff formula and that the resulting groups form the almost-Cross variety Cp. Cp, p > 3

Locally finite varieties are studied also in [164, 291, 508, 699, 971]. Free Groups. B. Neumann (1966) made the followlng conjecture: Any automorphlcally admissible subgroup of a free group of infinite rank is endomorphically admissible. Cohen (1968) proved the validity of an analogous assertion for free groups of infinite ranks in varieties of Abelian-by-nilpotentgroups. Neumann's conjecture is false in the general case; this has been established in independent papers by Bryant [394] and Ol'shanskli [165]. Bryant constructed a counterexample in the variety

(BdAN2)A~AC

, where the variety C is defined by

the identity [Ix,, x2], Ix3, Xd], Is," x6|i. Ol'shanskil proved the noncoincidence of the automorphic and endomorphic closures of element ~-----Ix~,x~#', X~x'|

in a free group with basis x1, x2, x3 .....

A new proof of the Magnus Frelheitssatz is contained in [406], and of its strengthening, in

678

[871].

Refinements of the bound for the rank of the intersects groups of a free group were obtained in [416, 618].

of flnltely-generated sub-

An example of an endomorphically admissible subsemlgroup of a free group, not being a subgroup, is given in [652]. A property is indicated in [700], characterizing all possible counterexamples (the first one of them has been pointed out by Dunwoody) to the weakened formulation of the Auslander--Lyndon theorem: if S~Pn, R>F~, S'~R', n~2, then S < R (in the theorem S ~ F ~ ). Automorphisms of Free Groups. Riggins and Lyndon [579] indicated a new method for investigating the automorphism groups of a free group. This method led them to obtain a sufficiently simple algebraic proof of the well-known Whitehead theorem (1963) on the automorphic conjugacy of the elements of a free group, proved by him by topological methods. A presentation, different from that of Nielsen, of the automorphlsm group of a free group in terms of generators and defining relations is obtained by McCool [719], which too is based on the ideas of Higglns and Lyndon. Also see [721]. The automorphlsm groups of a free group are studied in [600] by using the concept of the length function. Let U = { U h U 2 ..... um} be a collection of cyclically irreducible words of a free group Fn. McCool [720] examines the automorphlsm group AU of Fn, stabilizing U to within cyclic reduction. His main result is the finite presentation of group A U. The proof is effected by constructing a finite connected 2-complex whose fundamental group is isomorphic to AU, It is remarked that this same result can be obtained for the usual stabilizers too. The stabilizers of certain commutators of a free group have been computed in [407]. The papers of Dyer and Formanek are devoted to proofs of the perfection of certain automorphism groups. In [463] this is established for the group AutFn: in [464], for the automorphism group of a free 2-nilpotent group of rank m # l, 3. A more general result on the perfection of the automorphism group AutF/R' is proved in [462] under the fulfillment of certain conditions on R. Dyer and Scott [465] have obtained certain Structure theorems connected with the action of a finite automorphism group on a free group. Periodic automorphisms of free group F~ are studied by Miskin [737]. Other Questions. The nilpotency of classes of finite-rank free groups in products of varieties of primary exponents are computed in [242]. Wreath products of Abelian groups are investigated in [102, 210]. In [209] Sushchanskii computed the Engel length e of variety A~, having proved that e is strictly less than the nilpotency class of a rank 2 free group in this variety. Also see [886, 941]. The minimal ranks of free groups generating certain varieties are computed in [525, 526, 532, 778, 930]. Classes of groups, connected with varieties, are analyzed in [395, 542, 544,

576]. Decompositions into nontrivial direc~ products of finite free groups in certain varieties of groups are made by Houghton [604]. Bryant [388, 389] studies critical groups of varieties. Verbal subgroups of groups are examined in [19, 20~ 970]~ In [44] Golovin and Bronshtein systematize the accumulated material on exact operations. Limanskli [135] proves an analog of the Remak--Schmidt theorem for the case of a nilpotent product. Novak [157, 158] studies solvable products. 4.

Solvable Groups

The circle of papers on solvable groups and their generalizations is exceedingly wide. Here we can possibly consider only certain fundamental avenues laid out by the investigators in this area. See Robinson's monograph [832] for the other directions which relate mainly to various generalizations of the concept of solvability. Also see the lectures of Baumslag [312] and Warfield [942] o n nilpotent groups. Of importance is the short but astonishingly powerful report by Kargapolov at the Second International Conference on Group Theory in 1973 [629]. In it he presents the results of the Novosibirsk group-theory school originated by Kargapolov. In addition, in the same place he noted the interesting problems in the theory of solvable groups, the majority of which are as

679

yet unsolved. This report serves as an excellent example of how Kargapolov was able to focus attention on urgent important problems and of how he could inspire not only himself but others as well to solving them. Finiteness Conditions. Robinson's meaningful survey [838] included historical remarks on the development of the theory of groups with finiteness conditions on Abellan subgroups and also outlined the contemporary state of this theory. Researches on solvable groups with different finiteness conditions began in the forties with the works of Shmidt and Chernikov. Shmidt (1945), under the influence of a well-knowntheorem of Chernikov, proved that a solvable group with the minimal condition for Abelian subgroups is a finite extension of a direct sum of a finite number of quasi-cyclic groups (Chernlkov groups). Chernikov (1951) generallzed this statement to locally solvable groups. Mal'tsev (1951) proved that solvable groups with the maximal condition for Abelian subgroups are polycyclic. Kargapolov (1962) established that solvable groups with Abelian subgroups of finite ranks themselves have finite rank. Finally, in 1963 Kargapolov proved that a locally finite group all of whose Abelian subgroups are finite is itself finite. This resultwas obtained independently by Hall and by Kulatilaka (1963). Gorchakov (1964) examined locally solvable groups, establishing the finiteness of the rank of a torsion group with Abelian subgroups of finite rank; Merzlyakov (1964) too examined them, proving that if the ranks of the Abellan subgroups of a locally solvable group are bounded from above by number n, then the group itself has a rank no greater than f(n) for some function f having positive integer values. For locally solvable torsion-free groups finite generation of the Abelian subgroups does not imply finiteness of the rank of the whole group; an appropriate example was constructed also by Merzlyakov (1969). We remark that Kargapolov's 1962 theorem was called "the most profound result in solvable groups" in survey [838]. The papers of Adyan and Shunkov (see Sec. 5) are close to this theme. Subsequently, the theory of groups with different conditions on the subgroups was developed by Chernlkov's school. In their papers Chernlkov and his students examined groups in which not only some finiteness conditions or others are imposed on the subgroups, but also various conditions for the complementability of subgroups (many interesting results here were obtained by Zaitsev), conditions for the normality of subgroups, etc. It was established, for example, that an infinite locally solvable group with the minimal condition for noninvariant Abelian subgroups contains an infinite Abelian normal subgroup of finite index. A large part of these papers, including surveys, was published in transactions [64-66, i00, i01, 156]. Also see the survey articles [81, 235, 237]. Gruenberg [511] discusses Hall's ideas in connection with the analysis of solvable groups whose group rings are Noether. Shmel'kin [247] applied ring theory methods for investigating groups; also see [11, 12]. Beer (1965) proved that a solvable group with the minimal condition for normal subgroups is torsion. The existence of an unsolvable locally-solvable torsion-free group not satisfying the minimal condition for normal subgroups has been established in [570]. MacLane (1956) constructed an analogous example in the torsion case. Groups with the minimal condition for normal subgroups were investigated also by Silcock (see the literature cited). A survey on this subject was made by Wilson [972]. In 1969 Heineken and Mohamed constructed an example of a group with a normalizer condition and a trivial center. Various generalizations of this result are contained in [223, 565, 568, 569, 664]. Also see [554]. Zaitsev [79] proved that for locally-solvable groups the minimal conditions for subgroups and for "solvable subgroups of a given degree of solvability" are equivalent. Groups Close to Free Ones. The classical Magnus Freiheitssatzfor one-relator groups is very well known. Romanovskii [190] established a Freiheltssetz for groups prescribed by one defining relation in varieties of solvable and nilpotent groups of given length. Let us state one of his results.

Let F (i) denote the i-th commutant of a free solvable group F with basis

x, yl, ..., Ym' m>~2.

680

Let r be an element of F(k-I)/F (k), taken in some notation in terms

of the basis elements and let r be obtained by the deletion of x in this notation~

If (and

only if) r is not conjugate with r modulo F (k) , then the elements y~, ...~ Ym generate modulo the relation r = ! a free solvable group of the same desired length as F. An analogous Freiheitssatz was proved by Yabanzhi [262] in varieties NcA , c ~ 2

o

Lyndon posed the following problem: Let a group G be prescribed by the generators x ~ ..., xm and by n defining relations9 where m > n .

From xl, .o., xm can m -- n elements be

chosen which would generate in G a free group and would constitute its basis?

Romanovskii

[191] gave an affirmative answer to Lyndon's question in the variety of all groups and in the varietiesA n, n~1. The following question remains open: Question 3.

Is Lyndon's conjecture true in varieties N o , c ~ 2 ?

A complemented subgroup of a free group of variety V is called projective in variety V, Hall (1954) conjectured that all projective groups in varieties of zero exponent are free and established it for nilpotent varieties. Hall's conjecture proved to be false in the general case, as follows from the results of P. Neumann (1967) and of Ol'shanskii (1968). However, the question of its validity for various "natural" varieties of groups is of interest~ Relying on the posi=ive solution to Serre's problem on the freeness of finitely-generated projective modules over the polynomial ring, Artamonov [17] confirmed Hail~s conjecture for proJective metabelian groups of finite rank. The situation is different in the varieties AAn~ where for any n > 2 we can find examples of nonfree projective groups [288~ 730]~ The structure of not finitely-generated projective metabelian groups has not been investigated as yet, No answer has been obtained to: Question 4.

Is Hall's conjecture true in varieties

A~,n~3, AN~, N~A,c>/2?

In [827] Khemtulla showed that for any finitely-generated 3-solvable group G a number d exists such that every element of a com~mutant of G is a product of no Bore than d commutators (the number d is called the width of the oommutant of G). Since this result is useful when studying a cormnutant in the topological completions of G, the following question is of definite interest: Question 5. group finite?

Is the width of a commutant of an arbitrary finltely-generated solvable

Completions of metabelian groups were studied in detail by Kuz'min [120j !21]; also see [655]. Outside of the class of metabelian groups the comDletions of solvable groups have been investigated very little. Embedding Theorems. The majority of the theorems of the preceding paragraph were proved with the aid of the Magnus embedding of a free solvable group, the essence of which is the following. Let Fn(A k) be a free solvable group of exponen~ k and rank n and T be a left free module of rank n over ring Z(Fn(A~)).'" Then the free solvable group Fn(Ak+l) can be embedded in the group M of matrices of the form ( ~

) , where fEFn(Ak), t~T.

Group M can be understood dill_ ferently as the wreath product of a free Abelian group of rank n and of Stoup Fn(Am)~ Group (Ak+lFn ) is "well embedded" in M since its elements in M are picked out by a simple linear equation; see [183] for details on this. Thanks to the Magnus embedding many questions concerning solvable groups are reduced to the corresponding questions for modules over group rings of groups with a lower solvability exponent. In particular, a number of algorithmic problems for metabelian groups reduce to problems for free modules over ring Z[xl• ..... x~ ~I] , Using a new method Baumslag [315, 320] and Remes!ennlkov [180, 182] proved an analog of Higman's theorem in the variety of metabelian groups: Any finltely-generated metabelian group can be embedded in a finitely-presented group. Boler [558] refined this result for finiterank metabelian groups, showing that the large group also can be taken to be finlte-rank metabelian. Thomson [916] has shown that a flnltely-generated solvable linear group can be

681

embedded in a flnitely-presented solvable linear group. He noted as well that the group constructed in [181] satisfies the maximal condition for normal subgroups, so that the Hall problem on the existence of flnitely-presented solvable groups without the maximal condition for normal subgroups remains open. % A finltely-generated solvable group has been constructed in [449], not satisfying the maximal condition for normal subgroups, all whose central factors are finitely-generated; this solves another problem of P. HalI. Remalningopen are: Question 6. Can any flnltely-generated Abelian-by-nilpotent group be embedded in a finitely-presented Abelian-by-nilpotent group? Question 7. solvable group?

Can any group finitely presented in An be embedded in a finitely-presented

Hall (1954) proved the embeddability of an arbitrary countable group in a 2-generator 3solvable group. B. Neumann and H. Neumann (1959), using the construction of a wreath product, established that a countable d-solvable group is embeddable in a 2-generator (d + 2)-solvable group. Similar embedding theorems are useful when answering various questions in group theory. Roman'kov [193, 194] obtained assertions of the same type: i) a finitely-generated nilpotent group is embeddable in a 2-generator nilpotent group; 2) a polycyclic group is embeddable in a 2-generator polycyclic group. His methods of proof differ from those used previously. They are based on the representation of the groups being studied by matrices. It is well known that in the class of all countable groups there does not exist a universal group, i.e., a group in which is embeddable any group of the class given. Kargapolov [104] proved an analogous result for varieties NcA, c ~ 2 , and for countable orderable groups. Automorphisms. With each group G we can connect a sequence of automorphism groups A~G = AutG, AaG = AutA~G, Ai+~G = AutAiG. If G is a center-free group, then AIG can be embedded in Ai+~G. We can define A=G in a natural manner for any ordinal =; we obtain a tower of automorphism groups. By Wielandt's theorem a sequence of automorphism groups of a finite group can be stabilized at a finite step. In [825] this theorem is generalized to Chernikov groups. Hulse [609] proved that an automorphism tower of a polycyclic group is stabilized at a countable ordinal. He himself carried this result over to solvable As-groups in the Mal'tsev classification. The well-known Baumslag conjecture on the periodicity, starting at some stage, of the sequence of automorphism groups of any finitely-generated nilpotent group is to some extent confirmed by the Dyer--Formanek result mentioned in Sac. 3 that the automorphism group of a free 2-nilpotent group of finite rank (# i, 3) is perfect (for rank 3 groups the second group in the sequence of automorphism groups is perfect). In this connection we state the open Question 8. Will the automorphism group of an arbitrary free c-nilpotent group of sufficiently large rank (e.g., of rank greater than c + i) be perfect? Goryaga [60] showed that the automorphism group of a free c-nilpotent group of rank n>/32e-2+e, c>/2 , is a c-generator group. Bachmuth and Mochizuki [300] conjectured that for any finitely-generated automorphism groups of a finitely-generated solvable group the conclusion of Tits' well-known theorem on linear groups is valid: it either contains a free non-Abelian subgroup or is almost solvable. Roman'kov [195] established the embedding of finitely-generated groups O~ of a specific type in the automorphism groups of finitely-generated solvable groups H= . Among the groups G~ there is a wreath product T~ of any finite group C~ with an infinite cyclic group. If in this wreath product the finite group C~ is unsolvable, then the conclusion of Tits' theorem is false for it, viz., a counterexample to the Bachmuth-Mochlzuki conjecture. Another such example has been constructed independently by Hartley [561]. From Koman'kov's example we extract a larger one: First, we can take the solvable group H~ to be torslon-free, second, if the finite group C, is simple (andnon-Abelian), then the wreath product T, does not have a finite subnormal series whose factors either are Abelian or are representable by matrices over a Noetherlan commutative ring; this is an answer to another question from [300]. The following problem remains open: which countable groups are embeddable in the automorphism

*Added in proof. I

Abels (preprint) proved that such a property is possessed by the group of

* $

matrices (

9 0

682

1

) over ring Z(i/p).

groups of finitely-generated solvable groups? constructed.

Examples of unembeddable groups have not been

Questions often arise on carrying the known properties of linear groups over to matrix groups over a division ring. Thus, at the Second International Conference on Group Theory Bachmuth asked a similar question relative to Tits' theorem. So did P!otkin (Kourov Notebook, 2.64), relative to the coincidence of the set of Engel elements of a matrix group with its locally nilpotent radical. In connection with this we cite a few of Roman'kov's remarks. The well-kno~m Golod's construction (regarding it see the Kargapolov-Merzlyakov book Foundations, for example) can be modified so that as a result we obtain a 2-generator Engellan nonnilpotent orderable torsion-free group G. As is well know.m, the group ring of an orderable group is embeddable in a division ring; therefore~ G is a matrix group over a division ring. It is easy to see that the conjectures at the beginning of the paragraph are invalid for it. Groups G and H are commensurable if in them there are isomorphic subgroups of finite index G0~Ho . Baumslag [312] showed that commensurable flnltely-genera=ed nilpotent groups have commensurable automorphism groups. The converse is false; see [160] on this~ In [296) Bachmuth, Formanek, and Mochizukl proved that any autamorphism, identity modulo a commutant~ of a free rank 2 solvable group (or a group of type F/R' with specific conditions on R) is an inner automorphism. This theorem and several fragmentary facts are all that is known at the present time about the automorphism groups of free solvable groups~ Helneken and Liebeck [567] established that any finite group G is rep~esentabie in the form G = AutH/AutcH , where H is a 2-nilpotent group of period p 2 Autc H is the group of automorphlsms of H, identity modulo its center. Liebeck [682] generalized this theorem by proving that group H with the same condition can be chosen in the class of 2-nilpotent torslon-free groups. The well-known theorem of Gasch~tz on the existence of an outer automorphi~ for a finite noncyclic p-group has been generalized by Zalesskli [83] to nilpotent p-groups. Also see [908]. At the same time the theorem is false for arbitrary torsion-free nilpotent groups

[84]. Nilpotent groups in which the automorphisms (endomorphlsms) of any subgroup are extended to the whole group were studied by Kirkinskil [109]. Structure Theorems. The following result of Gasch~tz is well known in the theory of finite groups: The complementability of the Sylow subgroups of an Abelian normal subgroup A of a group G in the Sylow subgroups of group G implies the cemplementability of A in G. Zaitsev [80] generalized this result to the infinite case. Questions on the complementability of subgroups were examined by Newell (see the literature cited). In particular, in [766] he proved that if A ~ G , A is Abelian, G/A is nilpotent, and A satisfies the minimal condition, then A has a nilpotent complement N in G, which has a finite index in its normalizer. For any other such complement M there exists a finite subgroup B of A, normal in G, such that BN and BM are conjugate. * Frattini subgroups and series have been examined in a number of papers [205, 662, 664, 665, 673]. Erzakova and Churkin [74] constructed a 2-generator 4-solvable torsien-free group whose Frattini subgroup is nonnilpotent. An example of a solvable torsion group with such a condition is due to Hall. Sesekin [204] proved that in a group G = A B , where the factors are Absllan and at least one of them is finitely generated, we can find a nontrivial subgroup N contained in A or B. Recently Zaitsev obtained an analogous statement for locally cyclic torslon-free subgroups A and B. Also see [275, 276, 648]. Permutable subgroups are studied by Lennox [669, 674]. Qther Questions. Robinson [831] proved that a finitely-generated group possessing an ascending normal series with Abelian or finite factors either is itself nilpotent or possesses a nonnilpotent finite homomorphic image. Lennox [663] established that in the class of finitely-generated groups possessing ascending normal series, all of whose factors are almost Abelian~ if all 2--generator subgroups of a group G are polycyclic (almost nilpotent, supersolvable, nilpotent-by-finite~ nilpotent, 9Added in proof. Also see: D. I. Zaitsev, "Solvable groups of finite rank," Algebra Logika~ I~6, No. 3, 303-312 (1977).

683

finite), then the group G itself is such. In a special case this result occurs in [907]. The analogous assertion for the property of being metabelian is false [516]. See [564, 627, 762, 913] on Engel groups. Yakovlev [263, 264] examines lattice isomorphlsms of solvable groups, proving that the solvability property is preserved under a lattice isomorphism; however, a torslon-free solvable group, in general, is not determined by the lattice of its subgroups. Let a group G have the generator set X={I, x ~ , .... x~l}. Then we can define a growth function 7x(m)=IXm[ . In 1968 Milnor posed the problem: is it true that every finitelygenerated group has either a poly~omlal or an exponential growth? It is well known that a nilpotent-by-finite group has a polynomial growth, whereas all other almost-solvable groups have an exponential growth. On this see [308, 807, 846]. 5.

Torsion Groups

Burnside Groups. In 1968 Novikov and Adyan proved the existence of an infinite group with two generators and with the identity relation xn = 1 for any odd exponent n~>4381 , having by the same token solved the well-known Burnside problem. By Bm(n) we denote a free group of rank m in a Burnside variety of period n. In [6] Adyan gives a simplified proof of the infiniteness of the group Bm(n) , m ~ 2 , which was used to lower the original bound on the odd period n to n/>665 . In the same paper a number of more subtle properties of groups Bm(n) were obtained under the same restrictions: I) the centralizer of any nonunlt element is cyclic -- a solution of Kargapolov's problem on the existence of an infinite group, all Abelian subgroups of which are finite; 2) group B3(n) is embeddable in group B~(n); 3) for composite n the group Bm(n) does not satisfy the maximal and minimal conditions for normal subgroups; 4) group Bm(n) has an exponential growth (see Sec. 4 for the definition). In [245] Shirvanyan generalized statement 2), having established the embedding B~ (n) +B=(n) 9 The method worked out when solving the Burnside problem permitted the answering of a number of other question in group theory, including questions on nonperiodic groups. Thus, an example is constructed in [4] of a finitely-generated torslon-free group in which any two cyclic subgroups intersect trivially (the noncommutative analog of the additive group of rational numbers). This group is a central extension of an infinite cyclic group by group Bm(n). An independent relation system for Bm(n) has been indicated by Shirvanyan [246]. Another application of the method occurs in Adyan [7] in which the concept is introduced of n-perlodic products of groups for n ~ 6 6 5 . In the class of involution-free groups this product turns out to be an associative exact operation possessing the property of heredity with respect to subgroups. By the same token a partial affirmative answer has been obtained to the question on the existence of such operations other than direct and free products. In 1973 Britton [379] also published a proof of the theorem on the infiniteness of group Bm(n), m~>2 , for sufficiently large odd n. However, omissions were detected in his proof

(see [6, p. 6]). For a period 2k, k ~ , Burnside's problem remains open. The following question by Shunkov (Kourov Notebook, 4.74) is of interest in this connection: Question 9.

Does an infinite simple 2-group exist?

New examples of infinite finitely-generated torsion groups were constructed by Aleshin [9]. Engel Groups.

A group G is called m-Engel if the identity

[x, 9,~Y,....y ] = l

is fulfilled

m

on it. It is easy to see that 2-Engel groups are 3-nilpotent. Heineken (1961) proved that 3-Engel groups are locally nilpotent. In the general case it is very difficult to answer Question i0.

Will an arbitrary m-Engel group be locally nilpotent?

Heineken proved also that a 3-Engel group without elements of orders 2 and 5 are nilpotent. The restriction on the orders is essential since a nonnilpotent 3-Engel group of period 4 does exist (e.g., see the Kargapolov-Merzlyakov book'Foundations, p. 167). On the other hand, Gupta proved that any 3-Engel 2-group is solvable [521].

684

A strong result was obtained by Razmyslov [177]: a (p -- 2)-Engel group K(p, p -- 2) of period p ~ 5 and of infinite rank, free in the Kostrikin variety, is not nilpotent. It is well known that the properties of solvability and nilpotency for groups of prime period are equivalent; therefore, the group K(p, p -- 2), p ~ 5 , is unsolvable. The result presented for the case p = 5 has been obtained independently by Bachmuth and Mochizuki [30!]. At the beginning of the fifties Higman and Hall made a conjecture on the solvability of the Burnside group B~(4) . Wright (1961) proved that the nilpotency class of group Bn(4) does not exceed 3n -- I. As a matter of fact, as Gupta and Newman [529] showed, the nilpotency class of group Bn(4) does not exceed 3n -- 2 for any e ~ 2 . From the results of a series of papers [528, 533, 529] by Gupta and others it follows that the unsolvability of group B~(4) is equivalent to the exact equality to 3n -- 2 of the nilpotency class of any group Bn(4), n ~ 2 . Many authors have published on the Hall--Higman problem (see the literature cited); however, these papers do not contain any essential advances towards its solution, Razmyslov [178] gave the best possible answer: group B~(4) is unsolvable~ As already noted, hence it follows that the nilpotency class of group Bn(4), n ~ 2 , equals 3n -- 2. Paper [178] contains as well a short proof of the unsolvability of groups B~(p k) for k = 2 if p > 2 and for k = 3 if p = 2. Locall[ Finite G r o u _ ~ The most brilliant achievement in this area is a positive solution to the minimality problem for locally finite groups: Every locally finite group with a minimal condition for Abelian subgroups is a Chernikov group. This result [250] completes a large cycle of papers by Shunkov. A modified account is given by Kegel and Wehrfritz [641]. An essential role in the proof is played by =he Shunkov characterization of group PSLs(k) over an infinite locally-finite field k of odd characteristic. If k is quadratically closed, then PSLs(k) contains infinite 2-subgroups and is characterized by the embedding properties of a single subgroup. An infinite proper subgroup H of group G is said to be 2-inflnite!y isolated in G if it contains at least one involution and if for every involution i%H with infinite centralizer we have CH(i) = CG(i). Let G be a simple locally finite group containing a 2-infinitely isolated subgroup H with an infinite 2-subgroup and, moreover, let H have an involution with a finite centralizer. Then a locally finite quadratically closed field k exists such that G=PSL2(k) and H is a centralizer of an involution in G. If, however, field k is not quadratically closed, then group PSLa(k) does not contain infinite 2-subgroups and is characterized by the embedding properties of a whole family of subgroups. To be precise, let group G be a simple locally finite group with a finite Sylow 2-subgroup, and, moreover, for any involution: the index of a maximum normal 2'-subgroup OCG(i) in CG(i) is finite. Further, let a nonemp=y set S of infinite Abelian subgroups of group G exist such that: (a) Xg~ E

for all

X~E, gs

,

(b) if i is an involution from No(X), Xs is finite,

and C o ( i ) ~ N o O 0

(c) for every involution i from G there exists a group XGE Then closed.

O~PSL2(k)

, then the centralizer Cx(i) such that Co(1)~No()0.

for some locally finite field k of odd characteristic, not quadratically

The following theorem is proved by using this characterization. Let G be a locally finite group satisfying the minimal condition for p-subgroups for all primes p. If the centralizer of any involution from G is a Chernikov group, then G is almost locally solvable, We sketch the proof. By Chernikov's theorem any Sylow p-subgroup P of group G is a Chernlkov group. If A is a minimal subgroup of finite index in P, then the pair (rank A, IP:AI) is called the p-dlmension of group G. The p-dlmensions of different groups are lexicographlcally ordered. Let us consider all examples of G, contradicting the theorem and having a minimal 2dimension. It =urns out that a simple group G necessarily exists among them. By ~ we denote =he set of all maximal divisible Abellan subgroups of group G. Set ~ satisfies all the condltions of the above-stated characterization theorem, so =hat if the Sylow 2-subgroups of group G are finite, then G ~ P S L 2 for some locally finite field of odd characteristic p. Bur then G obviously does no= satisfy the minimal condition for Abelian p-subgroups, Therefore, G must contain infinite 2-subgroups. If S is a Sylow 2-subgroup of group G, then the center of group S is nontrlvial. If z is a central involution from S, then X~E exists such that S<.~.C~(z)<.~ NG(X) = H 9 Subgroup H turns out to be 2-infinitely isolated. In addition, in H there exists an involution i with a finite centralizer CH(i). Thus, the characterization theorem is

685

applicable and, hence, G = P S L 2 for some locally finite field of odd characteristic, which again contradicts the minimal condition. We return to Shunkov's theorem. Simple groups exist among locally finite groups of minimal 2-dlmension with the minimal condition for Abelian groups but not for Chernlkov groups. We denote the set of such simple groups by ~ . We assume that ~ is not empty. A contradiction is obtained if we can prove that a group G6~ exists, all centralizers of whose involutions are Chernlkov. In each group G 6 ~ we construct an infinite strictly ascending sequence {Mn}n~N of perfect subgroups such that (a) A4~/Z(A4n)~Cn6~ , (b) the 2-components Tn of the center Z(Mn) group M n are finite and strictly increasing. It can be proved that the 2-dlmensions of groups Mn and Gn coincide and hence are equal to each other. It is well known that for any locally finite group H the orders of the p-components of group H'nZ(H) depend only on the p-dimenslon of group H. In particular, the orders of the 2-components Tn of group Z ( M n ) = (Mn)'nZ(Mn) are bounded, which contradicts the construction. The properties of torsion groups as a function of the properties of the centralizers of involutions are investigated in a number of papers. Thus in [253] it is proved that a torsion group possessing involutions with a finite centralizer is locally finite and almost solvable. If in a= infinite group G an element of prime order generates, together with its conjugate, a finite subgroup, then in G exists an element of finite order with an infinite centralizer [255]. Let H be an almost locally solvable torsion group containing an elementary Abelian subgroup L of order 4. If the centralizer in H of any involution from L is a Chernikov group, then H too is a Chernikov group [241]. Shunkov [252] proved that a locally finite group whose Abelian subgroups have finite ranks is a finite extension of a locally infinite group; in particular, by a theorem of Strunkov (1964) it has a finite rank. The following generalization of the well-known Kargapolov--Hall--Kulatilaka theorem was obtained in [254]: An infinite conjugately biprimitively finite group possesses an infinite Abelian subgroup. Shunkov's results were generalized to wider classes of groups. A group G is said to be (conjugately) biprlmitively finite if for any finite subgroup K of it, in the factor gorup NG(K)/K any two (conjugate) elements of like prime order generate a finite subgroup. Shunkov [249] posed the question: Will a (conjugately) biprimitively finite group with the minimal condition for Abelian subgroups be locally finite? He himself obtained an affirmative answer in [249] for p-groups, in particular, for 2-groups, while Ostylovskli [168] did so for groups without involutions. Sozutov and Shunkov [207] proved that a biprimitlvely finite group, all of whose infinite proper subgroups are Abelian, is locally finite. Noskov [161] weakened the Abellanness condition to the condition of 2-nilpotency. A wide literature is devoted to Sylow theory and to the theory of formations in locally finite groups (see the literature cited). Locally normal groups, i.e., torsion groups with finite classes of conjugate elements, were given much attention in a recently issued monograph of Gorchakov [53]. We state some results. Gorchakov [50] proved that a residually finite locally normal group is the homomorphlc image of a subdirect product of finite groups. A commutant of a residually finite locally normal group is a subdirect product of finite groups (Tomkinson, see [53] for the proof). A factor group by the center of a residually finite locally normal group also is a subdirect product of finite groups (Gorchakov [52], Tomkinson). 6.

Residually Finite Groups

The systematic study of residually finite groups, RF-groups for short, started at the end of the fifties. It turned out that finitely-generated RF-groups possess a number of specific properties: Hopficity, solvability of the equality problem for finitely presented groups, the automorphism groups of such groups are RF-groups as well. In addition, a profinite topology can be defined on any RF-group by taking all normal subgroups of finite index as the base for neighborhoods of the identity. When investigating the properties of a RFgroup G we often have to examine its completion G in the profinite topology: the proflnite group G Is a good topological object; a compact group. Natural subclasses of RF-groups are picked out in the following manner. Let p be some predicate defined on an arbitrary group. We say that group G is residually finite relative 686

predicate P, a RF0-group for short, if for any collection of elements not occurring in relation p there exists a homomorphism of G onto a finite group, such that the images of the elements being examined also do not occur in relation O, This concept is connected with aigorithmic problems in the following way. For example, let us consider a conJugacy predicate: RFC-groups. Let group G be finitely presented in a variety V. Let V be specified by a finite system of idenci~ies and assume that G is a RFC-group. We note that in this case the conJugacy problem is solved positively in G. The latter signifies that the set of word pairs (a, b) such that a is conjugate with b in G is recursive. It is obvious that this set is recursively enumerable. Therefore, it is sufficient to understand that its complement is recurslvely enumerable as well. Indeed, by virtue of residual finiteness relative to conJugacy the complement can be enumerated by sorting through all possible homomorphlsms of G onto finite groups. to

At the present time we can indicate four main trends in the theory of RF-groups. 1. The proof of residual finiteness relative to different predicates for concrete groups (for classes of groups). 2. Residual properties of concrete groups (of classes of groups) for subclasses of finite groups (nilpotent, p-groups, simple). 3. The genus problem, 4. The theory of profini=e groups. RF-Groups. Hall (1959) proved the finiteness of all subdlrect indecomposable finitelygenerated Abelian-by-nilpotent groups. From this it follows that all finitely-generated Abelian-by-nilpotent groups are RF-groups. At the same time Hall conjectured that his results could be carried over t o the wider class of Abellan-by-polycyclic groups. He pointed out the principal difficulties in generalizing the results and indicated a proof of the conjecture on the finite dimensionallty of irreducible representations of polycyclic groups over finite fields. This conjecture by Hall was confirmed by Roseblade [840]. Relying on Roseblade's theorem, Jategaonkar [619] proved thatany flnltely-generated Abelian-by-polycyclic group is a RF-group; the theorems noted are the most profound results in this area of group theory. The example of a finitely-presented eenter-by-metabelian group that is not a RF-group, constructed by Baumslag [321], is erroneous since his group proved to be Abelian-by-polycyclic and, therefore, a RF-group. An interesting complement to Hall's articles (1954, 1959) is that by C. Gupta, N. Gupta, and Rhemtulla [519]. The results in this article clarify the boundary between solvable varieties in which all finitely-generated groups are RF-groups and those in which this is not so. By a theorem of Gruenberg (1957) a discrete wreath product A wr B of RF-groups is a RFgroup if and only if either B is finite or A is Abellan. Mamuchishvili [141] generalized this theorem to a k-nilpotent wreath product for a finitely-generated group A. The statement of his theorem differs from that of Gruenberg's only in the last words "or A is k-nilpotent." Partial results on the residual properties of free products with amalgamated subgroup and of HNN-extensions have been obtained by Baumslag and Tretkoff [336], Boler and Evans [359], Dyer [459], Evans [479, 480], Gregorac [501], and Tretkoff [921, 922]. Questions on the residual finiteness of one-relator groups occupied Gildehuys [493], Pride [813, 815], and Meskin [735]. Baumslag [313] showed that a finitely-generated free-by-cyclic group is a RF-groupo There remains open Question ii.

Is any knot group a RF-group?

This problem was reduced in [359] to knot groups of a special kind. We pass on to RFC-groups. A new, more contemporary proof of Remeslennikov's theorem (1969) on the fact that any polycyclic-by-finlte group is a RFC-group has been obtained by Formanek [488]. This theorem has been generalized by Grunewald and Segal [514] who showed that if the images of subgroups A and B are conjugate in all finite factor groups for G, then A and B are conjugate in G. The main llnk in the proof of their theorem is the following result. Let G be an almost solvable group and M be a G-module whose additive group is finitely genera=ed. Let ~:G--~A4 be a derivation. Then the image of ~ is closed in The profinite topology of M. From this it is eady to obtain the assertion on the closedness of the G-orbit of M, a result playing a definite role in the proofs of preceding authors. In the paper Grunewald and Segal discussed the action on M also of unsolvable groups. In this case the G-orbits are not always closed; but if G is an arithmetic subgroup of GL(np Z), acting naturally on Zn, then the closure of any of its orbits is the union of a finite number of

687

orbits. Whether this is true for finitely-generated groups closed in the profinite topology of GL(n, Z) is an open question. The fact that free groups are RFC-groups has been established by many authors; see [184, 900, 949]. The preservation of the property of being a RFC-group under free products was proved independently by Remeslennikov (1969) and Stabs [900]. Remeslennikov [184] found necessary and sufficient conditions under whose fulfillment a discrete wreath product of RFCgroups remains a RFC-group. In the same place it was shown that groups SL(n, Z) are not RFCgroups; an independent proof was given by Matveev and Platonov (1970). In [183] Remeslennikov and Sokolov proved that free solvable groups are RFC-groups. Kargapolov, Timoshenko [105], and Wehrfitz [950, 954] constructed simple examples of finitelygenerated metabelian groups that are not RFC-groups. The fact that Fuchs groups are RFC-groups was proved by Stabs [902]. Stebe's papers [903, 904] concerning the residual properties of finitely-generated nilpotent groups are interesting for the methods used in the proofs. An open question is Question 12.

Will a finite extension of a RFC-group always be a RFC-group?

Residual Properties of Simple and Nilpotent Groups. In a brief survey of problems and results connected with RF-groups, Magnus (1969) noted the importance of research on free groups that are residually finite simple. The residual properties of a rank 2 free group F~ for certain subclasses of groups PSL(2, q) were found by Peluso (1966)~ Poss [812], and Katz and Magnus (1969). In the last paper a question was posed on the residual properties of F2 for the class of groups PSL(2, pn), where p is a fixed prime, and for the class of groups PSL(3, p), where p ranges over all primes. Gorchakov and Levchuk [54] proved the residual property of F2 for any infinite set of groups PSL(2, q), whence follows the answer to the first question. An affirmative answer to the second question was obtained by Levchuk [133]. The results in [54, 132, 133] recapitulate Levchuk's theorem: a free non-Abelian group is residually any infinite set of Semisimple non-Abelian factors of groups GL(3, q) and of Suzuki groups Sz(q). By the same token we have a partial answer to Gorchakov's question: can group Fa be residually any infinite collection of finite simple and non-Abelian groups? From an affirmative answer to this question would follow an affirmative answer to a similar question (Kourov Notebook, 3.10; H. Neumann, Varieties of Groups, Springer-Verlag, Berlin-Heidelberg--New York (1967), Problem 23): Question 13. simple groups?

Can the variety of all groups be generated by any infinite set of finite

Heineken and P. Neumann (1967) showed that the last question is answered affirmatively in the class of groups PSL(2, q) and Sz(q), and, consequently, in the class of all known finite simple groups. Levchuk's theorem covers this result and confirms an assumption advanced by Meskin: A free group can be residually any class of two-generator groups, which generates the variety of all groups. Let us pass onto groups that can be residually nilpotent and residually finite p-groups. The presence of such residual properties in a group enables us to use the methods of Lie rings when investigating its properties, and, by passing to completions, to use the methods of local algebra. Several papers have been devoted to answering the question: In which classes of groups does a group G being residually a torsion-free nilpotent group follow from G being residually a finite p-group for an infinite set of primes p. Kuz'min [125] gave an affirmative answer in the case of fini=ely-generated metabelian groups. Hartley [557] generalized this result to finitely-generated Abelian-by-polycyclic groups; see [364] as well. Noskov [159] and Segal [874, 875], relying on the results of Hall and Jategaonkar, showed that a finitely-generated Abelian-by-polycyclic torsion-free group is almost a residually finite p-group for almost all primes p. An interesting connection between the residual property and orderabillty of a group was discovered by Rhem=ulla [828]. He showed that if a group is residually a finite p-group for an infinite set of primes p, then it is orderable. In an unpublished paper Kargapolov proved even more, having constructed for such a group a central system with torsion-free factors; see [143] for the proof.

688

Kuz'min [125] found necessary and sufficient conditions for a finitely-generated metabelian group to be residually a nilpotent~ a torsion-free nilpoten=, and a finite p-group. Some of these conditions are formulated in the language of modules over a group ring, others in the language of the uniqueness of the solutions of certain equations,

R4F

Certain remarks are made in [68!, 747] on groups of the form F/JR, R], F is a free group~ , being residually nilpotent. The Genus Problem.

groups.

For a group G the symbol ~(G) denotes the set of all its factor

We say that finltely-generated RF-groups G and H are of the same genus if ~ ( G ) = ~ ( H ) .

The question on the cardinality genus problem for G.

u(Q)

of the set of isomorphism classes in a genus also is a

The concept of genus can be approached also from a somewhat different

point of view: If flnltely-generated RF-groups G and H belong to the same genus~ then their completions

~ and ~

in the profinlte topology are isomorphic.

inverse spectra for S and H by O={GF; $~}

and H = { H v ; $$}

To prove this we denote the

, respectively~ where Gv and

are factor groups of G and H by verbal subgroups corresponding to a locally finite variety V; ~ , $~

are the corresponding canonic homomorphisms.

Let

Isom(Q, H)={|som(G~ Hv); 8~} be an inverse spectrumt where I som(Gv~ ~ ) and O~ is the inducing operator. isomorphism between O

is a finite set of isomorphisms from G V into

Any string from the limit set for X:som(G~ H) defines an

and ~f.

G ~(G) =| for any Abellan group G.

This is not so for nilpo~ent groups.

The simplest

examples of nonisomorphic nilpotent groups G and H of the s~me genus were constructed by Baumslag [323]: G =

(a, b; a~5=I, b-lab=a6), H;= (c, d; c2s=l, d-lcd=cn).

nilpotent groups was investigated by Pickel [795-797].

The genus problem for

Relying on Boral's theorem on the

finiteness of the number of double eosets for the adele groups of a ~inear algebraic group defined over Q, he proved the finiteness of =(G) G.

for any finitely-generated nilpotent group

Let us remark on the algorithmic aspect of this result: from it we can easily derive the

solvability of the isomorphism problem for a fixed finitely-generated nilpotent gorup. behavior of function

=(G) for nilpotent groups was investigaKed by Harary and Piekel [540],

Mislin [747], Hilton and Mislin [591], and Lemaire [660]~ Question 14.

The

Is the number

Pickel's theorem raises

~(G) finite for any polycyclic-by-finlte

group G? *

A certain approach to solving this problem is given in [799]. In contrast to the nilpotent case there exist finig~ly-presented metabelian groups for which the cardlnality =(G) is infinite [798]. Therefore, the isomorphism problem for a fixed metabellan group cannot~ in the general case, be solved by the method of approximations. In connection with the isomorphism problem for a free (free solvable) group we pose QuesKion 15.

Is =(0) = 1 for any finitely-generated free (free solvable) group G?

In [800] Pickel showed that the genus of any group does not contain any proper factor group and that the question has an affirmative answer for groups free in a certain nilpotent variety.

*Added in proof. Grunewald, Pickel, and Segal (preprint) proved the finiteness of ~(G) for any polycyclic-by-flnlte group G. [Supplied by translator: F. J. Grunewald, P. F. Pickelj and D. Segal, "Finiteness theorems for polycycllc groups~" Bull. Am. Math. Soc. (N.S.), ~, No. 3, 575-578 (1979).]

689

If we restrict ourselves only to finite nilpotsnt groups, then byanalogy with the concept of genus we arrive at the concept of nilgenus. The nilgenus of a noncyclic free (free solvable) group contains more than one group; Baumslag (1967, 1969). In connection with this Baumslag introduced the notion of a parafree group; this is a group P that is residually nilpotent and for which there exist a free group F and an embedding ~:P-+P inducing the isomorphisms ~i:F/F~-+P/P~ modulo the terms of the lower central series. Properties of parefree groups were investigated by Baumslag and Stammbach in [333, 334]. When trying to answer Question 15 it is useful to first ascertain whether a group belonging to the genus of a free (free solvable) group is not a parafree group, which is not a trivial fact. Profinite Groups. The inverse limit of finite groups is called a profinite group. The inverse limit of finite p-groups is called a pro-p-group. A topological characteristic of profinite groups is the following: A topological group is a profinite group if and only if it is compact and fully unconnected. The topological space of aprofinite group is homeomorphic to the generalized Cantor discontinuum D ~ for a suitable choice of cardinality ~. Profinite groups play a definite role in investigations on the Galois theory of infinite-dimensional algebraic extensions, in the theory of compact analytic p-groups, and in arithmetic and geometric questions connected with localizations. The majority of results on profinite groups known to date concern pro-p-groups: i. Results on the cohomological dimension of pro-p-groups and of the Poincar~ group (see: J.-P. Serre, Cohomologie Galoisienne, Lect. Notes Math., Vol. 5, Springer-Verlag, Berlin-Heidelberg--New York (1965)). 2. Generators and relators of pro-p-groups (see: H. Koch, Galoissche Theorie der pErweiterungen, Springer-Verlag, Berlin--New York (1970)). 3. Theory of compact analytic p-groups (see: J.-P. Serre, Lie Algebras and Lie Groups, W. A. Benjamin, New York--Amsterdam (1965); [385]. Results on profinite groups are difficult to survey, separating them from conjectures, and we note only Andoskii's paper [10] and the answer to question 3.40 from the Kourov Notebooks, obtained by Mel'nikov [145]. 7.

Algorithmic Problems

The classical algorithmic problems are: the word problem, the conJugacy problem, the occurrence problem in a subgroup, the isomorphism problem brought into group theory from topology. They were first formulated by Dehn at the beginning of the century. In the midfifties, it was shown that they have negative solutions in the class of all groups. Therefore, at the present time these problems and some others are posed and solved for concrete groups (classes of groups). In addition, a great deal of attention is paid to a more refined analysis of the nature of the solvability or insolvability of a given algorithmic problem. The modern state of the theory is rather completely reflected in the proceedings of the California (1973) and Oxford (1976) International Conferences on the Algorithmic Aspects of Algebra (mainly group theory) and in the book [742]; also see [361]. Solvable Groups. Two approaches to the problems posed are possible in the class of solvable groups. The first is that the algorithmic problems are examined for a finitelypresented group (set of groups) under the condition that such a group proves to be solvable with a given solvability class, i.e., it is assumed that the solvability condition ensues from the definingrelations. Groups finitely presented in a given variety are studied in the second approach. Stressing the immense value of Higman's theorem on the embedding of a recursively-presented group in a finitely-presented one when solving many algorithmic problems, we begin by asking Question 16. Is Higman's theorem true in variety An , i.e., can any group recursively presented in A n be embedded in a group finitely presented in An? Question 17. Can any group recursively presented in NcA be embedded in a group finitely presented in NcA? The answers to these questions are affirmative when n = 2, c = I, since Baumslag [320] and Remeslennikov [182] have established that any finitely-generated metabelian group can be embedded in a finitely-presented metabelian group. 690

We pass on to the word problem in An . By a theorem of Hall a finitely-generated metabelian group is residually finite; therefore, the word problem for groups finitely presented in variety A a is solved positively. A direct algorithm for the word problem in A ~ was indicated by Timoshenko [213]. On the other hand, Remeslennlkov's theorem [185] asserts that when n ~ 5 a group finitely presented in A n exists in which the word problem is solved negatively; more precisely, it is shown how a concrete representation of such a group can be written out for each n ~ 5 . Among the unsolved questions on the word problem in solvable varieties we note Question 18.

The word problem in varieties NcA, c > 2

o

The results obtained very recently by G. P. Kukin on the word problem in solvable varieties of Lie algebras permit us to hope that flnltely-presented groups in NcA , c ~ 2 , with an unsolvable word problem will be constructed. Therefore, the boundary between solvability and unsolvability of the word problem should be sought somewhere around center-by-metabellan groups. Question 19.

The word problem in the variety of center-by-metabelian groups.

Even less is known about the conJugacy problem in A n. From a theorem of Remeslennikov it follows that when n ~ 5 there exist examples of groups finitely presented in variety An, for which it is solved negatively. On the other hand, Sarkisyan [200] indicated an algorit~ for solving the conjugacy problem for free polynilpotent groups, while Boler [357] did llkewise for a concrete class of metabellan groups. There are two fundamental results on the occurrence problem. On the one hand, Romanovskii [189] proved that it can be solved positively for metabelian groups. On the other hand, a method was indicated in [185] for constructing agroup finitely-presented in varietyAn, n ~ 4 , and a finitely-generated subgroup in it, in which the occurrence problem is solved negatively. Apparently, Romanovskii's theorem is valid also for finltely-generated Abe!ian-by-polycyclic groups. An interesting paper on the problem's statement and its solution is by Kopytov [ll6]. On the basis of Remeslennikov's theorem [185] a group G finitely presented in Anis constructed in [ii0] for each n ~ 7 , such that an algorithm does not exist which ascertains whether or not any group finitely presented in A n is isomorphic to G. Here an interesting unsolved problem is Question 20.

The isomorphism problem for metabellan groups.

We recall that the isomorphism problem for the identity group was solved negatively in the class of all groups. In the variety An the analogous assertion is false since therein G @ 1 if and only if G/G' # 1 and, consequently, the problem is reduced to Abelian groups for which, as is very well known, it is solved positively. We can also prove by induction on t h e solvability class that the property of a group to be finite or nilpotent is algorithmlcally recognizable in A n. Of interest here is finding in A n the boundary separating algorithmically recognizable properties from the unrecognizable ones. ~ne following questions asked by Kargapolov remain open and await solution: Question 21.

The occurrence problem for a free solvable group of class e ~ 3 .

Question 22.

The word problem for one-relator groups in variety An~ n ~ 3

.

Th e Isomorphism Problem for Nilpotent Groups. The isomorphism problem remains the main unsolved algorithmic problem for nilpotent groups. We note that the epimorphlsm problem similar to it in statement is unsolvable in varieties Nc, c ~ 2 (Remeslennikov [187])~ From a number of serious results of an algorithmic nature for linear algebraic groups, recently obtained by Sarklsyan, follows a positive solution of the isomorphism problem for finitely-generated nilpotent groups modulo the realizability of the "Hesse principle" for simply-connected semlsimple algebraic groups defined over the rational number field. The author kindly sent us the manuscript of his article and, therefore, in v l e w o f the importance of the result, we present a brief sketch of the proof of his theorem. Let G and H be isomorphic finitely-generated nilpotent groups. Then their Mal'tsev completions Go and ~ also are isomorphic, which, by a well-known theorem of Mal'tsev, is equivalent to-the isdmorphlsm of the corresponding Lie Q-algebras L and M. By resolving the question on the isomorphism of L and M, we can take it that these algebras are isomorphic over the complex number field since the elementary theory of this field is solvable. In this

691

case we can apply the theory of non-Abelian cohomologies (see: J.-P. Serre, Cohomologle Galoisienne, cited on p. 690), and algebra M is isomorphic_to algebra L if and only if the cocycle constructed with respect to M from the set H ~ (Gal (Q/Q), Aut L) is trivial. In one of his articles Sarkisyan proved that the triviality of the cocycle can be algorithmically recognized under the condition of realizabillty of the "Hesse principle" for slmply-connected semisimple algebraic groups defined over Q. Therefore, we can take it that L = M and that G and H are commensurable subgroups in the associated group of Lie algebra L. Since the theory of the p-adic number field ~ with a valuation predicate is solvable, it is rather simple to verify algorlthmically whether or not the Zp-completions of groups G and H are isomorphic for all primes p. From the commensurability of G and H it follows that the actual test is needed only for a finite number of primes p. If OzpgeHzp for some p, then G=~/-f , and 'the algorithm ceases.

However, if Q~/-/zp

for all p, then in this case a double

coset is constructed with respect to H by an effective method in the adele group of the group Aut L, and the question on the isomorphism between G and H is reduced to the question on the triviality of this coset. The latter, as Sarkisyan showed (once again modulo the realizability of the "Hasse principle"), can be verified algorithmlcally. In conclusion we note that the realizability of the "Hesse principle" has been proved for all simply-connected semisimple algebraic groups except group Es. For any answer to the question on the mystery of group Es there remains unanswered Question 23.

The isomorphism problem for polycyclic groups.

Unsolvable Groups. A new proof of the Novikov--Boone theorem on the unsolvability of the word problem is given in [730]. Collins [433] constructs a finitely-presented group, the word problem for which has arbitrarily specified recursively enumerable tabular insolvability degrees; see [360, 982] on the insolvability degrees of the word problem in other hierarchies. The irreducibility of the algorithmic word, power, and order problems one to the other has been proved in [434]. Valley [9281% refines the Higman embedding theorem by proving that with respect to a recursively presented group G we can so choose its enveloping finitely-presented group G* that the word problem in G* is solved with the aid of some Turing machine with oracle, where the machine works over a word of length k for no more than k 7 log k cycles and interrogates the oracle on the words of G, whose lengths do not exceed k. The connection between the word problem in semigroup ~ and in group r with the same relators were investigated by Sarkisyan [199]. It turned out that the solvability of the word problem in F does not follow from the solvability of the word problem in N even under the condition that ~ is embeddable in F . However, if the system of defining relations for N does not contain loops, then the solvability of the divisibility problem in ~ is sufficient for the solvability of the word problem in F. Meskln [736] constructs an example of a recursively-presented residually-finite group with unsolvable word problem. The main unanswered question on the word problem is Question 24.

The word problem for two-relator groups.

An unexpected result on the conJugacy problem was obtained by Goryaga and Kirkinskii [59] and independently by Collins and Miller [436]. They constructed a finitely-presented group with an unsolvable conjugacy problem and indicated in it a subgroup of index 2 with a solvable conjugacy problem. By the same token they have proved that in contrast to the word problem the property of a group to have a solvable conJugacy problem does not carry over to finite extensions. The conJugacy problem for free products with amalgamation occupied Bezverkhnii [24, 29, 30], Larsen [653], Hurwith [617], Clapham [425], and Lipschutz [690], while for HNN-extensions, Anshel [283] and Anshel and Stebe [285]. Moldavanskii [150] indicated an algorithm for the recognition of the conJugacy of subgroups in the free product A*B under the condition that such is the case in A and B and that the occurrence problem is solvable in A and B. The $A reading of this paper failed to reveal any mention of a Turing machine or, indeed, of any fact mentioned in this sentence except that Valley is concerned with the Higman embedding theorem. The reviews of this paper, both in Mathematical Reviews and in Referativnyi Zhurnai, confirm this. Valiev has dealt with the refinement of Higman's theorem, with reference to the Turing machined in other papers -- Translator.

692

conjugacy problem for certain classes of one-relator groups has been solved in [67, 68, 222]. However, still open is Question 25.

The conjugacy problem in one-relator groups.

Also as yet unanswered is

Question 26.

The isomorphism problem for one-relator groups.

Attempts to solve it in [738, 801, 820, 843, 844] led only to partial successes. The occurrence problem was examined mainly for free products with an amalgamated subgroup (see the literature cited). See Sec. 2 on the solution of algorithmic problems in small cancellation groups. Equations in Groups. The general question by Tarski on the existence of an algorithm determining the solvability of an arbitrary equation in a free group remains open. We present the known facts. Lyndon (i960) considered equations with one unknown and proved that all solutions of such equations can be represented by using the so-called parametric words. Lorents (1968) obtained a definitive result. The set of solutions of an arbitrary system of equations with one unknown is specified by the values of a finite collection of words of the form abmc, where a, b, c are cons=ants and m is a parameter taking positive integer values. From the proof of the result cited follows an algorithm determining the solvability of such equation systems. An independent proof, containing a gap, was given in [287]. From the classical results of Nielsen and Whitehead follows an algorithm recognizing the solvability of coefficientless equations, i.e., equations of the form w(x, y) = a, where the left-hand side does not contain a constant and a is an element of a group, Wicks [960, 961] gave a new proof of the algorithmic solvability mentioned. He presented a general solution of such an equation parametrized by the elements of some automorphlsm group of a free group and he investigated this automorphism group. A stronger assertion was obtained by Khmelevskii [224, 225], constructing an algorithm for the solvability in a free group of systems of coefflcientless equations and of equations with separated variables of the form u(x) = v(y), where the words u and v can contain constants. We note that the solutions of equations in two or more variables, in general, cannot be represented by the values of a finite number of parametric words. An explicit example of such an equation was given by Appel (1969). Also see [71, 139] on the solvability of arbitrary equations in a free group. Let us note the most interesting properties of the solutions in a free group of equations of a specific form.

Lyndon [694] proved that the rank of a subgroup generated by the solutions

of an equation of form y n ..y~,

n > 1 , does not exceed m/2.

The prevlously-known result of m Baumslag and Steinberg says that if in the equation w(x:, ..., x n) = a , m > ! ~ the word w is neither a nontrivial power nor a primitive element in a free group

with basis x:, ~o.,

Xn, then the equation's solution g~, ..., gn together with the element a generates a subgroup whose rank does not exceed n -- i. In [698] it was shown that a solution of an equa ~ n n =ion of form xl...x k = a~...a~, n > 1 , exists if and only if m ~ k . An analogous statement was obtained for equations of form [xl, x2]~..[X2n-1, X=n] = =~...=~ in [473]: A solution exists if and only if

~>12~+I

.

The principal assertions in [472, 696] are obtained as

corollaries of the results cited. Quadratic equations, i.e., equations containing each unknown exactly twice, were studied in [474, 802, 803].

The ranks of the subgroups generated by the solutions were examined in

[69, 906]. Consider the free product T = G,(x) and let r(x) = a o x i ~ x i 2 element of it.

...x ina

be an arbitrary n The m o s t important problem in the theory of equations over an arbitrary

group is the determination of when =he equation r(x) = 1 has a solution in some overgroup

693

I-I>/G

.

Gerstenhaber and Rothaus (1963) showed that a solution always exists if G is a R

finite group and

l(r)-----~it-bA0 .

Levin (1962) proved the existence of solutions for the case

t=l

when

it>0, ~ - I , 2 .....n 9

Chopenko [239] proved that when Z(r) # 0 there exists

dependent of G) such that the equation r(x) m = 1 has a solution.

m>/l

(in-

Schiek [864] gave a solu-

tion for equations of a specific form with l(r) = 0. The solvability problem for a coefflcientless equation with n unknowns in a free group of some variety of rank no less than n can still be treated as a problem of endomorphic reducibility in this group. Roman'kov [196] obtained an unexpected result: the problem of endomorphlc reducibility in a free nilpotent group of countable rank of nilpotency class no less than 9 is algorithmlcally unsolvable. An analogous assertion is valid for free solvable groups of countable rank [197]. Equations in groups were analyzed also in [16, 238, 408, 685, 962]. 8.

Model-Theoretlr Methods

,Alsebraically Closed Groups. At the beginning of the fifties Scott introduced the concept of an algebraically closed group. A group G is said to be algebraically closed, a.c. for short, (existentially closed, e.r for short) if any finite system of equations (of equations and inequations) with parameters from G, having a solution in some extension of G, already has a solution in G itself. Scott showed that any group is embeddable in an a.c~ group of the same cardinality. B. Neumann (1952) proved that, first, any e.c. group is simple, and, second, the concepts of a.c. and e.c. groups coincide for nonunit groups. The complexity of the structure of an a.c. group was emphasized by the following theorem of B. Neumann [753]: any finltely-generated group having a solvable word problem is embeddable in any a.c. group. Maclntyre [703] proved the inverse of Neumann's theorem: if a finitely-generated group has an unsolvable word problem, then an a.c. group exists in which it is not embeddable. Thus, at first a group-theoretic criterion was obtained for the solvability of the word problem. Later, Boone and Higman [363] obtained another criterion: a flnltely-generated group has a solvable word problem if and only if it can be embedded in a simple group, and that, in its own turn, in a finitely-presented group. In this connection we note the unanswered Question 27. Can any finitely-generated group with a solvable word problem be embedded in a finitely-presented simple group? Belegradek [340, 341] obtained algebraic characterizations of recursive relations on finitely-generated groups; some of them are formulated in terms of a.c. groups. General considerations show that any two a.c. groups satisfy one and the same v ~ sentences. Bars asked the question: are any two a.r groups elementarily equivalent? Macintyre [704] gave a negative answer to this question; furthermore, in [35, 597, 981] it was shown that the number of elementary types of a.c. groups equals 2~0 . In all the papers on Bers' problem an essential role is played by the following fact noted by Macintyre [704]: certain predicates not formula predicates in the class of all groups are formula predicates in the class of a.c. groups. For example, the n-ary predicate "subgroup generated by elements x:, ..., xn in G is simple" is defined in the class of a.c. groups. The question on which predicates are formula predicates in the class of a.c. groups was studied by Belegradek [34, 35] and Trofimov [213]. A definitive answer to the question is in [38]. Miller (see [704]) showed that any a.c. group has a finitely-generated subgroup with an unsolvable word problem. Since any recursively-presented simple group has a solvable word problem, it follows hence that no a.c. group can be specified by a recurslve set of defining relations. In [704] it is shown that any countable a.c. group contains a proper isomorphic copy, and, thus, minimal a.c. groups do not exist. Nevertheless, no a.c. group can be embedded in its own finltely-generated subgroup.

694

Hall (1959) introduced the following concept: A locally finite group is said to be a universal locally finite (u.l.f.) group if any finite group is embeddable in it and any two isomorphic finite subgroups of it are conjugate in it. It turned out that u.l.f, groups are precisely groups a.c, in the class of locally finite groups. Hall proved that any infinite locally finite group is embeddable in a u.l.f, group of the same cardinality and that any two countable u.l.f, groups are isomorphic. In connection with this Kegel and Wehrfritz [641] posed the following question: are any two u.l.f, groups of one and the same uncountable cardinality isomorphic? Macintyre and Shelah [707] answered this question negatively, Using model-theoretic methods, they showed that there exist 2~ types of the isomorphism of u.l.f. groups of any uncountable cardina!ity ~. The connection between field theory and the theory of algebraically closed fields led to the concept of a model companion. Let T and T' be theories of one and the same first-order language. We say that T' is a model companion of T if T' is model complete, any model of T ~ is embeddable in a model of T, and any model of T is embeddable in a model of T'. Theory T has a model companion if and only if the class of its e.c. system is axiomatizable (in this case the theory of e.c. systems is a model companion). For example, a.c. fields are a model companion of fields; real-closed fields are a model companion of formally real fields; divisible torslon-free Abelian groups are a model companion of torsion-free Abellan groups~ Eklof and Sabbagh [478] noted that the class of a.c. groups is not axlomatizable andj consequently, group theory does not have a model companion. When studying e.c. systems of theories the paper is arranged along the following plan. At first the question on the presence of a model companion is studied. If the answer is no, the question is studied on the difference between the e.c. systems obtained with the aid of forcing-constructions, the so-called finitely generic and infinitely generic systems introduced by Robinson. Saracino undertook the study by this plan of solvable and nilpotent groups. In [858, 859] it was proved that for any n ~ 2 the theories of n-solvable, n-nilpotent, and torsionfree n-nilpotent groups do not have model companions. In [858] it was shown that finitelygeneric metabelian groups are distinguished from infinitely-generic ones wi~h the help of v3v-sentences. In [859] an analogous result is obtained for n-nilpotent groups, n ~ 2 ~ and in [860], for torsion-free 2-nilpotent groups. Baumslag and Levin [331] studied a.c. groups in the class of torslon-free 2-nilpotent groups. It turned out that for any positive ~ < ~ there exists a unique countable group whose center is the product of m copies of Q. Saracino and Wood [861, 862] study periodic e.c. 2-nilpotent groups. They prove that these are precisely finitely-generic 2-nilpotent groups. Here only one countable periodic e.c. 2-nilpotent group exists. Any periodic 2-nilpotent group is embeddable in a periodic e.c. 2-nilpotent group. One countable e.c. group exists in any Burnside variety of 2-nilpotent groups, while the theory of this class has a countably categorical model companion. In these varieties the concept of an a.c. group is broader than ~he concept of an e.c. group. Saturated Abelian Groups. Complete theories of groups were actively studied in recent years. Complete theories of Abelian groups were described by Szmielew (1956). Eklof and Fisher [477] proved this result again, having classified saturated Abelian groups. We say that a model of A is saturated if for any subset X of A of cardinality less than the cardinallty of A the model of A realizes an arbitrary type F(x) of formulas with constants from X, which is consistent with a theory of the enriched model. We recall that every maximal noncontradictory set of a language's formulas dependent on a variable x is called a

type r ( x ) . It turned out chat the reduced part of a saturated Abellan group is the completion in the Z-adlc topology of a direct sum of a certain number of primary cyclic groups and of additive groups of p-adic primes. Since the Z-adic completion of an Abellan group A is defined uniquely (the inverse limit of groups A/hA, n~N ), saturated Abelian groups are classified to within isomorphism with the aid of Invariants that are cardinals. Hence, it is easy to obtain a classification of Abelian groups according to their elementary properties. Since any Abelian group is elementarily equivalent to a saturated group, from the classification of saturated groups we obtain a classification of all Abelian groups according to elementary properties. Let us present the elementary invariants from the article by Eklof and Fisher.

695

Let A be an arbitrary Abelian group and A[p] be its p-socle. Since dim(pkA/pk+iA) is mono=onically decreasing function of k, we can define the first elementary invarlant for A: limit value for { p~ A \ T, (p, A ) = | ifitisfinite, ,dim tp,--~a-i), oo otherwise.

Analogously, D (p, A)=

limit value for d i m if it is finite, oo otherwise .

(p~A[p]),

In addition, U ( p , n - - 1, A ) = {[ a- i m [p"-'A ~ / , [p]~ if it is finite, oo otherwise.

Exp(A)={~

ifAotherwiasebounded .is group,

Szmlelew's theorem [477] states: An Abellan group A is elementarily equivalent to an Abellan group B if and only if the elementary invariants of A equal the corresponding invariants of B. It would be interesting to study nilpotent saturated groups, and through this to classify them by the elementary properties. So far in this direction it is known only that from the elementary equivalence of finitely-generated 2-nilpotent groups does not follow their isomorphism (Zil'ber [87]). Categorical and Stable Grou~s. A number of properties of complete theories, proving important in model theory when studying categorlcity, were actively employed in recent years for classifying complete theories of groups. Let I be an infinite cardlnallty. A flrst-order theory is called categorical in cardinality I (or, in short, A-categorical) if all its models of cardinality A are isomorphic. If A is an algebraic system, X is a subset of its fundamental set, and ~, and a2 are its elements, then we say that a~ and ~a have one type over X, if a~ and aa satisfy in A one and the same formulas with constants from X. A complete theory is called A-stable if for any model A of it and for any subset X of cardlnallty ~A in A the number of types of the elements of system A over X does not exceed %. A theory is called stable if it is stable in even one cardinality. Morley (1965) proved that a countable complete theory categorical in some uncountable cardinality is categorical in all uncountable cardinalities and is M0-stable. A detailed discussion of the concepts presented can be found in [107, 198]. We say that a group is %-categorlcal or A-stable if its theory is such. Abelian groups categorical in uncountable cardinalltles, as well as M0-stable Abelian groups were described by Maclntyre [702]. It is easy to describe M0-categorical Abelien groups: they are groups of bounded exponent. Many authors have independently noted that the theory of any Abelian group is stable; see [88, 342], for example. The main unanswered question on the categoriclty problem is Question 28. groups?

What is the algebraic structure of ~0-categorical

and

M,-categorical

From the general characterization of M0 -categorical theories it follows that a countable group is Mo -categorical if and only if the number of classes of automorphically conjugate n-th elements is finite for any n. M 0 -categorical groups were studied in [483, 810, 811]. Zil'ber [89, 90] studied M,-categorical groups. In particular, he proved that an infinite group with an Ml-categorlcal universal theory is Abelian and that any infinite group categorical in all cardinalities is nilpotent-by-flnite. Baur, Cherlln, and Macintyre [337], and Felgner [484] independently, proved that any No-categorical stable group is nilpotent-by-flnite. In [337] it was proved that any M 0 categorlcal M0-stable group (in particular, any group categorical in all infinite cardlnalitles) is Abelian-by-flnite. All such groups (and even all No-categorical Abelian-byfinite groups) were described.

696

Belegradek (1972) proved that in any non-Abelian variety there is a nonstable group, and, consequently, by Shelah's theorem [880], 2 ~ groups of arbitrary uncountable cardlnality l (also see [305, 848, 849]). Groups show up unexpectedly in certain cases when trying to classify arbitrary categorical theories [i, 90, 171]. For instance, it turned out that the open question on the existence of finitely-axiomatizable MI-categorical but not ~0 -categorical quasi-varieties is equivalent to the fulfillment of one of two conditions: either an infinite finitely-presented ring exists, being a division ring, or an infinite finitely-presented group existsj in which there is a finite number of nonunlt conjugacy classes such that any nonunit cyclic group intersects one of them. Solvability of Elementary Theories. The main results on the solvability problem for elementary theories of groups (of classes of groups) are the unsolvability of an elementary theory of all groups (Tarskl), the unsolvability of an elementary theory of the class of all finite groups (Mal'tsev), and the solvability of an elementary theory of the class of all Abelian groups (Shmeleva). The main unanswered question is Question 29~

Is an elementary theory of a free non-Abelian group solvable?

Szmielew's theorem on the solvability of an elementary theory of the class of groups of any Abelian variety contrasts with Zamyatin's result [86]: every non-Abelian variety of groups has an unsolvable theory. This theorem serves as an answer to the well-known Tarski--Ershov question. Earlier Ershov [75] proved that if a variety of groups contains a finite non-Abelian group, then an elementary theory of this variety is unsolvable. Ershov [75] studied elementary theories of flnitely-generated nilpotent groups. He found that a theory of such a group is solvable if and only if the group is Abelian-by-finite. Using Robinson's theorem on the hereditary unsolvability of a theory of the ring of algebraic integers and the method of relative elementary definability proposed by Ershov (1965), Romanovskii extended the last result to polycyclic-by-finite groups (reported at the Fourteenth AllUnion Algebraic Conference, Novoslbirsk, 1977). An open question is Question 30: Will a finitely-generated solvable group with a solvable elementary theory be Abelian-by-finite? In [197] Roman'kov constructed an example of a finitely-generated metabelian 4-nilpotent group whose universal theory is unsolvable. Certain generalizations of Merzlyakov's theorem (1966) on the coincidence of positive theories of free noncommutatlve groups were obtained in [853]. However, still unanswered remains Question 31.

Will free non-Abelian groups of different ranks be elementarily equivalent?

We did not survey and did not cite in the literature numerous papers on elementary theories of Abelian groups with additional predicates, referring the reader to the survey [115] by Kokorin and Pinus. Miscellany. A group of results on infinite Abelian groups exists, provable under various set-theoretical assumptions (e.g., Shelah's theorem on the independence of Whitehead's conjecture in ZFC). Palyutin [172] studies the phenomenon of categoricity of a theory in the language L=..~ richer than the first-order language. A fundamental informal corollary of his results, in application to Abellan groups, is: it is impossible to classify uncountable p-groups to within isomorphism with the aid of reasonably defined invariants. 9.

Homological Methods

In the investigation of groups by the methods of homological algebra there naturally arise classes of groups distinguished by some finiteness constraints or others on the resolutions: groups of type (FP)n, groups of finite (co)homological dimension, groups of low dimensions, etc. The fundamental problem accompanying the distinguishing of new classes is to find their algebraic (not homological) characterization. At the present time a satisfactory solution to this problem has been obtained only under very rigid constraints on the resolutions. The difficulty of translating from the homological language to the usual language of group theory is, without doubt, an essential limitation on the possibilities of homologlcal methods.

697

Finitely-Generated Resolutions. RG-module.

Let G be a group, R be a commutative ring, and A be an

We say that A is of type (FP)n over R if a projective resolution ...--+Pr-+P0--+

A---~0 exists in which the modules Pi,

i ~ n, are finitely-generated.

generated for all i, then we say that A is of type {FP)= .

If they are finitely

Finally, A is of type (FP) if a

finite projective resolution 0--+Pn--+...---~P0--+A--+0 exists consisting of flnitely-generated modules.

Usually, a module A is of type (FP)o if and only if it is finitely generated and

is of type (FP) I if and only if it is finitely presented. (FP)n over R, where

,=oo

or an integer ~ 0 ,

Group G is called a group of type

if the trivial RG-module R has the type (FP)n.

If R = Z, then in this case G is called simply a group of type (FP)n. Question 32. tion.

Let group G be such that the G-molule Z has a finite projective resolu-

Can a finite free resolution be found for Z? As shown in [350], the class (FP)~ is exhausted by flnitely-generated groups.

Therein

it was proved that group G is of type (FP)a over R if and only if it is almost finitelypresented over R, i.e., G = F/N, where the module R| Question 33.

It is

is finitely-generated.

obvious that a flnitely-presented group is an almost finitely-

presented group over any ring R.

Is the converse true?

Bieri [350] and Brown [384] derive homological criteria for a group to belong to class (FP)n: A group G is of type (FP)n over R if and only if it is finitely generated and Hh(G, R R G ) = 0

for all

1~k~n

and for all direct products of

x=max(~0, IR[) copies of

X

ring RG.

Groups of type (FP)~

over R are precisely those groups for which the functor

Hk(G, --) or Hk(G, --) commutes with the direct limits for all k ~ 0 Commensurable groups have a like type [350]. the group N is of type

.

If in the exact sequencel--+N--~G--+Q--+l

(FP)~ over R, then groups G and Q are of like type.

Using a homological criterion and the Mayer--Vietoris exact sequence in a more general form for groups acting on trees, it has been obtained by Chiswell [421].

Bierl [350] proved

that a free product with amalgamation G = G~*sG2 with cofactors of type (FP)n over R and with an amalgamated subgroup of type (FP)n_ ~ over R is of type (FP)n over R.

An analogous result

is valid for HNN-extensions. Question 34.

Let C be =he smallest class of groups, containlng all finite groups,

closed relative to extensions, to the forming of free constructions with appropriate subgroups from C, and to the taking of subgroups of finite index. (FP)=~ not belonging to class C?

Do there exist groups of type

Is the torsion part of a group of type

(FP)= finite?

Do

all arithmetic groups SLn(Z) belong to class C (it is known [877] that they are of type (FP)| Group SLn(Z) is an "amalgam" of finite groups [891].* In [350] it was proved that if a group G has the type (FP)n, then the Abelian groups Hk(G , Z) and Hk(G, Z) for 0~.k~.n are finitely generated. As Bieri [350] showed, groups of type

(FP)~ have a specific feature: for them exist

spectral sequences approximating (co)homologies with coefficients in an arbitrary RG-module by means of (co)homologies with coefficients in RG. *As can be seen from its tltle, [891] dealt with the group SLs(Z). Also in [877] Serre apecifically proves that group SLs(Z) is not an amalgam-- Translator.

698

Homological Dimensions. n if K i = 0 for all i>n morphism

~:A--+B

smallest integer

.

An RG-reso!ution ...-+KI-+K0-+C--+0

We recall that a left RG-module M is called flat if for any mono-

of right FG-modules the map n>0

by definition has length

~|

I:A|174

also is a monomorphism,

The

for which there exists a flat (projective) resolution of module C of

length n is called the flat (projective) dimension of module C over R.

If as C we take the

trivial RG-module R, then we obtain, respectively, the homological hdRG and the cohomological cdRG dimension of group G.

The (co)homological dimension can also be defined as the smallest

n for which an RG-module A exists such that Hn(G, A) # 0 [Bm(G, A) # 0]. From the general facts on the relation between flatness and proJectivity it follows [350] that h d R O ~ c d a O

and

cdRG~-~hdRO+l if G is countable and hdRG = cdRG if G is of type

(FP)~

over R.

As yet little is known on the dependence of the dimensions of group G on the

ring.

We make several simple remarks on this.

If hdRG (or cdRG) is finite, then G does not

have R-torsion, i.e., the order of any element of G either is infinite or is invertlble in R. Further, cdRG = 0 if and only if G is a finite group without R-torsion~ hdRG = 0 if and only if G is a locally finite group without R-torsion. From the Lyndon--Hochschild--Serre spectral sequence I--~N--+G--+Q--+I

for group extensions

it follows immediately that the (co)homological dimension of group G does not exceed the sum of the dimensions of groups N and Q. is strict.

Many examples exist of groups for which the inequality

However, it becomes an equality in certain interesting oases.

Fel'dman [221]

showed that if N is of type (FP) over R and the R-module Hn(N, RN) is free when n = cdRN, then from cdaQ
it follows that CdRG = cdRN + CdRQ.

The assumption that N is of type (FP) is

essential (counterexample: G is a rank 2 free group and N = [G, G]).

However, it can be

shown [347] that membership in type (FP) can be replaced by the considerably weaker condition of the existence of a projective resolvent finitely generated in the upper dimension n = cdRN.

An analogous statement is valid for the homological dimension. Serre (1965) proved that if

torsion, then CdRS = cdRG.

S~.~G

is a subgroup of finite index and G does not have R-

The homologlcal analog is in [350].

Let us consider the behavior of the dimensions under the taking of free products and HNN-extensions.

From the Mayer-Vietoris sequence it follows [348] that if G = G~,sG2 , n =

max(cdRG, , odRG,) , then n~.~cdRO~_n+| . follows that CdRG: = CdRG~ = cdRS = n. gated subgroups

{S, o(S)} 9

Furthermore, from the equality cdRG = n + ! it By G = G~,S, ~ we denote the HNN-extension with conju-

If cdRG1 = n, then n < c d a G < n + l

equality cdRG = n + 1 it follows that cdRG: = cdRS = n. the homological dimension. groups acting on trees.

, and, moreover, from the

Analogous statements are true for

Gildenhuys [494] ascertained the cohomological dimension of

Let a group act without inversion on a tree X.

By V(X) we denote

the vertex set and by E(X) the edge set of tree X; Gp (respectively, G x) is the stabilizer of vertex p (respectively, of edge x).

The main result is that if

sup{cdxlx~E(X)} , then nv
nv=sup{cdO~Ip%V(X), hE= and nv<~cdG<~nv+l

As an application there is proved the Lyndon (1950) inequality e d g e 2

for groups with one relator r that is not

a

proper exponent.

For this the group is made to

act on some tree X in such a way that the stabilizers of its vertices admit of embedding in

699

another one-relator group, the relator having a length less than that of r. In another paper [495] Gildenhuys obtained a generalization of Lyndon's inequality. The dimensions of "constructable" groups were computed in [328]. In [349] it was shown that if G is a non-Abelian group of finite cohomological dimension and Z(G) is Its center, then c d Z ( G ) < c d O . Low Dimensions. It is easy to see that groups of zero cohomologicaldlmenslon over R are Just all finite groups without R-torsion. The problem of classifying groups of dimension ~-~] remains open, but it has been solved for torslon-free groups by Stallings and Swan (196869): A group G has a cohomological dimension ~.I over R if and only if it is free. If a group G can be written as G = G~,sGa, G I ~ S ~ G ~ ,

ISl
, then we shall say that G

has an s-decompositlon. However, if G is a HNN-extension with finite conjugate subgroups, then we shall say that G has a 0-decomposition. The proof of the Stalllngs--Swan theorem is based on the following fundamental result of Stallings (1968): Let G be a finitely-generated group and let H*(G, RG) # O, then G has either an u-decompositlon or a 0-decomposltion. (It is easy to see that cdmG~<|~H1(O, R G ) ~ O , so that the Stalllng~-Swan theorem for flnitelygenerated groups follows at once from this result and Gruchko's theorem. Swan generalized the theorem to infinitely-generated groups.) The theory of rings in groups played an important role in the proof of Stallings' theorem. Hopf and Freudenthal (1943) created the theory of the ends of a topological space. They even showed that if a group acts well on the space, then the structure of the ends of the space depends only on the group, so that the ends can be determined in the group itself. Let G be an infinite group. The module Z2G = Homz(ZG, Za) can be identified with the set of all subsets of group G relative to the symmetric difference operation; under such an identification ZaG is the set of all finite subsets of group G. Let ~ G = Z 2 G / Z 2 G 9 It is easy to see that dimH~ ~ ) = l + d i m H l ( O , Z2G) and this number is called the number of ends of group G. Stallings (1968)showed that a finltely-generated torsion-free group having infinitely many ends is free. The description of infinltely-generated groups with an infinite number of ends is a problem to which the papers [605, 606~ 608, 780] were devoted. Oxley [780] and Houghton [605] Independently showed that if a group is not locally finite, then it has i, 2, or infinitely many ends; moreover, G has two ends if and only if it contains a cyclic subgroup of finite index. Here abuts Sarkisyan's paper [201] wherein he computes the group H~(G, ZG) in the case when G is a countable locally-finite or solvable group. It is not difficult to show that a countable locally-flnlte group has Infinitely many ends. There is a conjecture that an arbitrary uncountable locally-finite group has only one end (Kourov Notebook, 5.61). For an arbitrary group G and an Abelian group A the first cohomology group HI(G, A | is determined by the number of ends of group G. Houghton [607] showed that the second cohomology group H 2 ( G , A | depends on an analogous invarlant, to be precise, on the socalled "group at infinity of group G." In many cases the latter coincides with the fundamental group of G (Johnson [621] and Lee and Raymond [656]). The papers of the latter authors have analogs in =he results of Farrell [481] and Bieri [349] on group H2(G,A| . A group G is called O-accessible (or u0-indecomposable) it has neither =-decomposltions nor 0-decompositions. We say that G is n-accesslble if G has an m-decomposition with (n -- 1)accessible cofactors or is a HNN-extension of an (n -- l)-accessible group. Accessibility signifies n-accessibility for some n. From Grushko's theorem it follows easily that any finitely-generated torsion-free group is accessible. It is not known whether this is true for arbitrary finitely-generated groups. A very nice accessibilltycrlterlon for almost finitely-presented groups has been obtained by Bamford and Dunwoody[350]: an almost finitelypresented group G is accessible if and only if H*(G, ZG) is finitely-generated as a G-module. Accessibility can also be expressed in the language of the theory of groups acting on trees (Sec. i). From the theorem on subgroups of free consgructions it follows that the accessibility of an arbitrary group G is equivalent to G being the fundamental group of a finite graph of groups with ~0-indecomposable vertices and finite edges. Let G be a finitely-generated accessible group. Then cdRO
700

The most well-known example of groups of cohomologi~al dimension 2 is, as Lyndon (1950) showed, that of torslon-free groups with a single defining relation. From the profound results of Papakyrlakopoulos (1957) it follows that the fundamental group of a nontrivial knot has a cohomologlcal dimension 2. However, it is not known whether every knot group has a one-relator presentation. The class of groups of eohomologlcal dimension 2 is considerably wider than the class of all subgroups of kn0t groups or one-relator groups, since it is closed rela~ tiveto free products with free amalgamated subgroups. There are simpler examples, viz.~ the direct product of two non-Abellan free groups has cohomologlcal dimension 2, but is embeddable either in a one-relator group or in a knot group. In [350] it was shown that a countable locally-free group has cohomologlcal dimension 2. A finitely-presented normal subgroup in a finitely-generated group with cd~.~2 either is free or has a finite index [349]. Finitely-presented groups with cd<2 , containing a nontrlvial finitely-generated free normal subgroup, are described in that same paper. For a group G = F/N, as Swan (1969) showed, c d G < 2 if and only if N/[N, N] is a ZG-projective module. Solvable and Nilpotent Groups. Fel'dman, generalizing the earlier results of Sta~mbach, proved that the homological dimension of a solvable torsion-free group G coincides with its Hirsch number h(G), while the inequality h(O).~
Duality Groups. We say that G is a Polncar4 duality group (of dimension n) if A) for all G-modules A and all k~Z , where these isomorphlsms are natural with respect to A and commute with the boundary homomorphlsms of the long exact sequences for H*(G, --) and H,(G, --). Such groups were first investigated (in the discrete case) by Bieri [345] and Johnson and Wall [623]. Group Z is the only Poincar4 duality group of dimension 1. Examples of Poincar~ duality groups of dimension 2 are the fundamental groups of two-dimensional closed surfaces of genus ~ | . Whether other examples exist is an open question~ Every flnltely-presented Poincar~ duality group of dimension 2 is residually nilpotent [350]. Every subgroup of a two-dimensional Poincar4 duality group is either locally free or of finite index [481, 482]. Polycyclic groups, and only they, are solvable Poincar~ duality groups [345]. If in the extension |--+N--+G--+Q--+| the groups N and Q are Poincar~ duality groups of dimensions m and n, then G is a Poincar4 duality group of dimension m + n [345].

Hn-k(O. A)~H~(G,

Group Extensions. Conditions are presented in [781] for tha splittability of a fixed extension G of a group N, in terms of the ideals of the semigroup of noncommutat~e cohomologics H*(G, N). In [782] Pandya and Bercov, using cross homomorphisms (l-cocycles) instead of 2-cohomologies, gave simpler proofs of a number of known existence and conJugacy conditions for the complements to Abelian normal subgroups in groups with operators. Wells [959] considered automorphisms of group extensions. If 1-+N--+G-"+Q --~1 is a group extension, then the pair (~,x)~AutN• {xI/~Q} Q--~Out N

is a system of coset

Q

is not necessarily induced by automorphlsms of group G.

representatives,

under which /--+x--+Aut N--~OutN

sistent if T leaves the subgroup

ker~

If

then by ~ we denote the homomorphism

for any x from Q.

The pair (~ z) is called con-

invariant and the automorphlsm induced by r on ~(Q~

701

coincides with the inner automorphism induced by the image ~.

By C we denote the group of

consistent pairs and by Aut(G, N) we define the group of automorphisms of group G that leave the subgroup N invariant.

The main result in [112] is the construction of the exact sequence

1--+ZI(Q, Z(N))---~Aut(G, N)---~C-+Hi (Q, Z(N)~ . 910] also are devoted to extensions.

Papers [151, 152, 409, 636, 783, 785, 847, 856,

Different kinds of exact sequences associated with group

extensions are constructed in [466-469, 535, 536,

577, 773, 774].

Hill [582] gave a geo-

metric interpretation of obstructions to the existence of extensions. Multiplicators.

The group H2(G, Z) is also named a Schur multlpllcator of group G.

When G coincides with its own commutant the multlplicator can be defined as the kernel of the universal central extension [F, F]I[R , F]

~G

of group G, where F is a free group and

F / R = G.

Baumslag [311] constructed an example of a finltely-generated not finitely-presented group with a trivial multiplicator (see [963] as well). Next he showed [325] that such examples exist in the class of finltely-generated not finltely-presented metabellan groups. If a group is specified by m generators and n relations in the variety of metabelian groups and m--n>l , then the group's multiplicator is not finitely generated and, hence, it is not finitely presented [327]. In [54] it was shown that if G is a splittable extension of group N by group Q, then the multlplicator of group Q is a direct cofactor in the multiplicator of group G. Multiplicators of verbal wreath products have been studied as well. Papers [65, 66, 220, 311, 317, 325, 327, 335, 470, 537, 625, 643, 905, 963] are devoted to multiplicators and their connections with K-theories. Books. First of all we note Gruenberg's book [510], presenting cohomologles as an instrument for the study of groups. The homological tools are presented in the least necessaryspace. A distinctive feature is the systematic use of a special resolution (now called the Gruenberg resolution) constructed from the group's generators and relations. In addition, almost everything that was known (up to that time) on the cohomologlcal dimension of discrete groups was collected in the book. New results on dimension and Polncar~ duality groups are contained in Bieri's book [350] devoted to the generalproblem of classlfying discrete groups by their cohomological properties. In his notes [427] Cohen gives a selfcontained exposition of Stallings' results on groups of cohomological dimension 4 1 (also see book [809]). Stammbach's book [898] is devoted to the use of (co)homologies in the study of group localizations and of the central series of groups free in varieties. The techniques of group cohomology theory, needed in class field theory, is presented in Weiss' book [958]. Babakhanian sets forth the cohomology theory of groups (mainly finite), and its application in group theory, in book [294]. One chapter is devoted to the Hochschild--Serre spectral sequence. The last two books are distinguished by their elementary nature and by the completeness of their exposition and, therefore, can be recommended for initial acquaintance with the subject. Miscellany. A number of papers (chiefly by Hilton) were devoted to p-adic completions and locallzations of nilpotent groups, interest in whlch has been motivated by the recent applications of these ideas in topology [585-589, 594-596]. In the papers [657-659] of Leedham-Green and [895, 897, 898] of Stammbach, homologieal methods are applied in the theory of varieties of groups. The Hochschild--Serre spectral sequence is studied in [592, 593, 934]. In an interesting paper [891] Soule computes the cohomologies of group SLs(Z).

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