Arch. Math., Vol. 55, 521-532 (1990)

0003-889X/90/5506-052t $ 3,90/0 9 1990 Birkh/iuser Verlag, Basel

Barely transitive permutation groups By M. Kuzucuo(~LU

1. Introduction. A group of permutations G of an infinite set ~2 is called a barely transitive group if G acts transitively on ~ and every orbit of every proper subgroup is finite~ An abstract group is called barely transitive, if it is isomorphic to some barely transitive permutation group. An infinite group G is barely transitive in its regular permutation representation if every proper subgroup of G is finite, so the quasicyclic group Cp= is a barely transitive group in its regular representation. Thus as is stated in [7] barely transitive groups are a natural generalization of the Schmidt groups, which are infinite groups all of whose proper subgroups are finite. The existence of non-abelian Schmidt groups was announced by Ol'sanskii in [14]: there is an infinite non-abelian group G in which every proper subgroup has prime order, and any two subgroups of the same order are conjugate. Hence Ol'sanskii groups are simple barely transitive groups on their regular representations. Examples of barely transitive, locally finite groups with G ~e G' :~ 1 are first given by Hartley in [6] and [7]. These examples are locally finite p-groups satisfying the normalizer condition, with trivial centre and such that every proper subgroup of G is subnormal and nilpotent. They are similar to the groups constructed by Heineken and Mohamed in [11]. In [6] G/G' ~- Cp~ and G' is an elementary abelian p-group. In [7] Hartley has constructed, for each natural number n a locally finite group G. satisfying, G,/G~ ~ Cv= such that G,~ is abelian like G' in [6], but unlike G', the group G~ has exponent p". Locally finite barely transitive groups with G #: G' are studied in [7] and the structure of these groups is reasonably well understood. In this work we will investigate the abstract properties of perfect locally finite barely transitive groups in the hope of eventually deciding whether such groups exist. Our first aim is to prove such a group must be a p-group, though we have not suceeded in doing that. By using similar techniques as in [1], [2], [3] we prove: Theorem 1.1. Let G be a locally finite, locally p-solvable barely transitive group containing a non-trivial element of order p. Then 1. G is a p-group. 2. Every proper normal subgroup is nilpotent of finite exponent. There are examples of infinite locally finite p-groups with G = G' and we do not know whether such a group can be barely transitive. Ol'sanskii groups are examples of barely

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transitive groups with G = G'. Clearly Ol'sanskii groups are not locally finite. We do not have examples of locally finite-(solvable) barely transitive groups with G = G'. The minimality condition, even on p-subgroups, seems a strong condition, since with such an assumption we can prove that Theorem 1.2. A locally finite barely transitive group containing a non-trivial element o f order p and satisfying min-p is isomorphic to Cp|

Next we turn our attention to simple groups. Using the classification of finite simple groups we prove Theorem 1.3. Let G be a locally finite simple group and suppose G = ~) Gi, where i=1 G 1 < G 2 < ... and the G i are finite simple. Then G is not barely transitive.

In fact we prove a more general result. In section 4 we define the notion of a semisimple element of a countable simple locally finite group, and note that if G is as in Theorem 1.3, then G contains semisimple elements. We prove Theorem 1.3'. Let G be a countable locally finite simple group containing a semisimple element. Then G is not barely transitive.

It remains unclear whether a locally finite barely transitive group can be simple. 2. Locally p-solvable groups. We state some well-known results and facts about barely transitive groups. The following is stated in [7]. Lemma 2.1. An infinite group G can be represented faithfully as a barely transitive group if and only if G possesses a subgroup H such that 0 Hx = 1 and IK : K c~ Hl < oo for every proper subgroup K < G. ~

In the sequel H will denote a point stabilizer of a barely transitive group. Lemma 2.2. Let G be a locally finite barely transitive group. i) G does not have a proper subgroup of finite index. ii) I f there exists a finite normal subgroup N of G, then N <=Z(G). iii) For any proper subgroup K of G, K H 4= G. P r o o f. Trivial. Lemma 2.3. Every abelian barely transitive permutation group is isomorphic to a quasicyclic group Cp~ for some prime p. Proof is an application of (2.1) Lemma 2.4. Let G be a locally finite barely transitive permutation group on a set 0 and N be a proper normal subgroup of G. I f ~ (HN/N) gs = N, then G/N is a barely transitive permutation group gN

on ~ where ~ is the set of all orbits of N.

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P r o o f. Since N is a p r o p e r normal subgroup, by definition of bare transitivity, the lengths of the orbits of N are finite and equal. Let 0 be the set of all orbits of N. Then GIN acts on O transitively and S" f i N : = S 9 g where S is an orbit of N. The length of an orbit of each proper subgroup K / N of G / N containing S is finite, as the orbit of K is finite when K is regarded as a group acting on O a n d the order of S is finite. It remains to show that the stabilizer of a point in 0 is ( H N / N ) ~ for some g N ~ G/N. Let H = G, and let S be the orbit of N containing ct. If g N ~ (GIN)s, then ct. g n = c~ for some n ~ N whence g n e H and g N ~ H N / N . Conversely, H N / N certainly stabilizes S. Hence (G/N)s = H N / N . It is clear that G / N acts transitively on O. So given S i e ~ there exists g N ~ G / N such that S " g N = Si (G/U)s, = (G/U)s. oN = (G/U)~ N = ( H N / N ) ~ So the stabilizer of points in O are ( H N / N ) ~ for some g N ~ G / N and _by assumption (~ ( H N / N y N = N. Hence by (2.1) GIN is a barely transitive permutation group on O. This will give oN

us a non-trivial barely transitive permutation g r o u p GIN provided that ~ ( H N / N ) ~ = N. ON

L e m m a 2.5. I f N is a maximal normal subgroup o f G, then G / N is a barely transitive permutation group on the set o f orbits o f N. P r o o f. This is immediate corollary of (2.4) We require some results based on Belyaev, [3]. The following is slightly stronger than the statement in [3]. L e m m a 2.6. Let X be a subgroup o f an arbitrary group G, and let g be an element o f G o f finite order n. I f I X : X n X g] is finite, then ]X ~ : X n X 91 is finite. The subgroup X ~ X ~ c~ . . . c~ X g" 1 is normalized by g and has finite index in X . P r o o f. We first show that if [X : X n Xgl < oo, then [X : X c~ Xg n . . . n

Xr

< oo

for all m > 2. We do n o t assume at the m o m e n t g has finite order n. We prove this by induction o n m. If m = 2, then it is true by the hypothesis. Assume that IX:X

n X ~ n . . . c~ Xg~-~]< oo,

then IX g : X ~ n X g~ n . . . n Xgm[ < oo. But this implies IX n X ~ : X c~ X ~ c~... n X~

< oo.

Hence [X:X n X ~ n...n

X~

= [ X : X c~ X~ IX n X ~

n X ~ c~...nX~

oo

as required. N o w if g has finite order n, then by the above observation I X : X c~ XO n . . . c ~ Xg"-~[< oo and X n X ~ n . . . c~ X 9"-' is normalized by g and IX~ : X c~ X~ < IX~ : X ~

g2 n . . . n

X ~

c~ X ] < oo.

Hence IX" : X n X ~ oo. The following is stated in [3]. F o r convenience we give the p r o o f here. L e m m a 2.7. Suppose G is a locally finite primitive permutation group on a set f2 and G~ is the stabilizer o f a point c~e (2. I f some non-trivial orbit o f the stabilizer G~ is finite, then G is finite.

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P r o o f. Since G is a primitive permutation group G~is a maximal subgroup of G. By assumption IG~ : G~,pl < ~ for some c~ 4: fl ~ O. But G~,a = G~ n Ga and by transitivity of G there exists, g e G such that Go = Gt~. Hence by (2.6) and Lemma 5 of [3] IG: G~I < [(G~, Ga) : G~ n Gp[< oo. Hence G is a finite permutation group. Lemma 2.8. A locally finite, barely transitive permutation group cannot be a primitive permutation

group. P r o o f. Follows from (2.7). Lemma 2.9. Every 2-transitive permutation group is primitive [15]. Therefore by (2.8) a locally finite barely transitive permutation group cannot be 2-transitive. Lemma 2.10. Let G be a locally finite barely transitive permutation group. Then G has

a tower of subgroups H < H1 < H 2 . . . such that G = 0 Hi and for every proper subgroup f=l

K of G, there exists n ~ N such that K < H n. Moreover any two proper subgroups of G generate a proper subgroup. P r o o f. We first show that if there exists a maximal subgroup M, then M does not contain H. For, assume if possible that M contains H. Then [ M : H I is finite. I M X : H X I = IM : H I < oo for all x e G. We have also [H x : H n H~[ < oo for all x e G ]H x-~ : H n H~-~[ = { H : H n H~[ is finite for all x e G. Hence

[M : H n H~[ = [M : H I I H : H n HX[ and IM x : H ~ H~[ = [M x : H~[ IH ~: H c~ HX[ are finite. Then by Lemma 5 of [3] ](M, M ~ ) : H n H ~'] < ~ for all x e G. Since H < M, we have [(M, M * ) : H [ < ~ for all x ~ G. But this implies for all x ~ G, the group ( M , M ~) is a proper subgroup of G. Maximality of M implies that M = M ~ for all x ~ G, hence M is a normal subgroup of G. Then the quotient group G/M does not have a proper non-trivial subgroup. Thus G/M is a finite group but by (2.2) G does not have a proper subgroup of finite index. Hence M can not contain H. By (2.8) G is not a primitive permutation group i.e. the stabilizer of a point H is not a maximal subgroup of G. Hence there exists xl ~ G - H such that H < H1 = (H, x~) < G. Bare transitivity of G implies [(H, x l ) : H[ < 00. Since by the above paragraph H is not contained in a maximal subgroup of G the group ( H , x~) is not a maximal subgroup of G, so there exists x 2 e G - H1 such that H1 < H 2 = (H~, x2) < G and IH2 : H [ < 00. C o n t i n u i n g in this way we find a tower of subgroups H < H 1 < H 2 . . . such that IH1 : HI < IH2 : HI < IH3 : HI < ... hence i~)~1Hi" n

is infinite. This implies G = 0 Hi. i=l

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N o w let K and L be two proper subgroups of G. Then [ K : K c~ H I < oo and IL: L c~ H I < ~ . Let {kt, k 2 . . . . . k,} be a transversal of K n H in K and {l~,..., Ira} be a transversal of L c ~ H

in L. Since G =

~)H~ there exists r e N

such that

i=1

{kx, k 2 , . . . , k,, 11. . . . . l,,} ~ Hr. But then ( k l , k2 . . . . . k,, l~ . . . . . l,,, H ) < H, since H, contains H. On the other hand (K, L ) ~ (k 1, k 2 .... , k,, 11. . . . . l~, H ) < Hr. Hence ( K , L ) < H, as required. In particular G does not have a maximal subgroup. L e m m a 2.11. A locally finite group of Lie type cannot be isomorphic to a barely transitive group. P r o o f. By [4] (Proposition 8.2.1) every Chevalley group has a (B, N)-pair, and if a group G has a (B, N)-pair then by [4] (8.2.2) G = BNB = (B, N). But by (2.10) a locally finite barely transitive permutation group cannot be generated by two proper subgroups. Hence G cannot be barely transitive. This of course also follows from Theorem 1.3 but the above argument is more elementary. L e m m a 2.12. Let G be a locally finite barely transitive permutation group. I f there exists a proper normal subgroup N containing H, then G ~ G' and GIN ~- Cvoofor some prime p. In particular if G = G', then H ~ = G. P r o o f. Let N be a proper normal subgroup of G containing H. By (2.2) IG : N] is infinite, but each proper subgroup K / N of G/N is finite. But by [5] every infinite locally finite grouP contains an infinite abelian subgroup. Hence G/N must be abelian. Therefore, G' < N. The group G/N is an abelian group in which every proper subgroup is finite. N o w it is easy to see that G/N is isomorphic to Cp~. Therefore if G = G', then H ~ = G. L e m m a 2.13. Let G be a locally finite barely transitive group. Then i) every proper subgroup of G is residually fnite. ii) Z(G) is a finite group and Z 2 ( G ) = Z(G) i.e. the hypercentre is equal to the centre. Proof. i) Let K be a proper subgroup of G. Then [ K : K c ~ H [ < ~ [ K X : K X n H l < o o for all x 6 G . Hence I K : K n H X - I [ < ~ for all x ~ G . /

0

and But

N

Kc~HX<=Kc~(OHX]=I.

So for all k ~ K

there exists K c ~ H r<=K with

xEG

k ~ K • H y and IK : K n H y] < oo. Hence we can find a normal subgroup in K • H y of finite index in K as required. ii) By bare transitivity [Z(G): Z(G) :n H I < oo. Clearly Z(G) n H-~ G. But by (2.l) H x = 1. Hence Z(G) ~ H = 1. This implies that Z(G) is a finite group. x~G

Let Zz(G ) be the second term of upper central series of G, and let z ~ Zz(G). Define

Q : G~--~Z(G) o ~

[o, z] "

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M. KUZUCUO~LU

Then ~ois a homomorphism. Since Z(G) is finite, G/Ker O is a finite group. But by (2.2) G does not have proper subgroups of finite index. Hence Ker Q = G which completes the proof of (2.13).

Lemma 2.14. I f G is a locally finite barely transitive group, then G has an infinite subset M which generates G and every infinite subset of M also generates G. Moreover G is countable. P r o o f. The index IG : H Lis infinite. Let M be a transversal of H in G. The set M is an infinite set and (H, M ) = G. Since by (2.10) any two proper subgroup of G generate a proper subgroup of G and H is a proper subgroup, ( M ) = G. N o w let B be any countably infinite subset of M, then ( B ) = G. Indeed by the same method as above [(H, B) : HI is infinite. Hence (H, B ) = G. Again by (2.10) we have ( B ) = G. Let B = {x 1, x 2 .... } and B~ = {x~ ... x~}. Then G = ~) (B~). Locally finiteness of G i=1

implies that B~ is finite for all i ~ N. Hence G is countable. P r o o f o f T h e o r e m 1.1. Let G be a locally finite locally p-solvable barely transitive permutation group, and let p be a fixed prime which divides the order of an element of G. By (2.14) G is countable and so G can be written as a union of finite subgroups G~ such that G = ~) Gi i=l

G1 < G2 < Ga < .... Since G1 is p-solvable, a Hall if-subgroup of G1 exists and any two of them are conjugate ([15] Theorem (9.1.6)). Construct inductively P, ~ SyleG. and Q, e Halle, G. such that P, < -P,+ 1, and Q,, < Q,,+ 1 with G,, = P~ Q,,. Since any two Hallp,-subgroups are conjugate, this is possible. Let

i=1

i=1

Since PQ contains all G~ and G = u G~, PQ = G. But G is barely transitive so by (2.10) either P = G or Q = G. But there exists an element whose order is divisible by p. Hence G = P and G is a p-group. For (2) we argue exactly as in the proof of the theorem in [7].

3. The minimality condition on barely transitive groups. P r o o f o f T h e o r e m 1 . 2 . Let G be a locally finite barely transitive group containing an element of order p and satisfying min-p. By [9] (Theorem B) a locally finite simple group satisfying min-p is of Lie type. But (2.11) implies that our group cannot be of Lie type. Hence a locally finite barely transitive group satisfying min-p cannot be simple. Let 1 4= N
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not have a proper subgroup of finite index. Hence G is a divisible abelian group. Then 2.3 implies that G ~- Cp~. Assume that G is not a p-group. Then P is a proper subgroup of G and by (2.13) each proper subgroup of G is residually finite. But Cp~ is not residually finite hence P must be a finite group. Since we have non-trivial normal subgroups, either G has a maximal normal subgroup or G is a union of an ascending series of proper normal subgroups N~. In the latter case there exists i such that P ~ N~< G and by the Frattini argument a = N, N~(e).

But G cannot be generated by two proper subgroups, and N~ is a proper subgroup so G = N~(P). Hence P is a normal subgroup of G. The group P is finite and normal whence (2.2) implies that P <=Z(G). Since P is finite abelian and a maximal p-subgroup, G/P is a if-group. Let Z be a local system consisting of finite subgroups and containing P. We can find such a local system since G is countable and P is finite. Any element K i in the local system is a finite subgroup of G containing P and ([Ki/P[, IP[) = I. Then by the Schur-Zassenhaus theorem Ki = P x L i as P < Z(G). The group Li is a f - g r o u p . But this is true for all K~ ~ S. Since the complements L~ of P are unique by embedding for each i, L~ < Li + a we get G = P • O<(G). Since P is finite and G does not have a subgroup of finite index. G = Op.(G) which is impossible since there exists a non-trivial element of order p. It remains to show the first possibility, that G contains a maximal normal subgroup is impossible. If there exists a maximal normal subgroup N, then GIN is a simple group satisfying min-p. By (2.5) GIN is barely transitive and by the first paragraph a barely transitive locally finite group satisfying min-p cannot be simple. This proof also says that in a locally finite barely transitive group all maximal p-subgroups are infinite and indeed not (~ernikov. One can prove the following statement without using the classification of finite simple groups. A locally finite barely transitive group containing an involution and satisfying min-2 is isomorphic to C2=. 4. Simple groups. First we give definitions about simple groups. Definition. Let G be a group. The sequence 2 ( = (Gi, Mi)iEN is called a Kegel sequence of G if G = w Gi, where Gi is a strictly ascending sequence of finite subgroups of G and M i is a maximal normal subgroup of G~ satisfying G~ ~ Mi+ 1 = I for all i. Definition. Let G be a countable locally finite, simple group, • = (Gi, M~)Z~N a Kegel sequence of G. The element g ~ G is semisimple if o(9) = n and there exists an infinite

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sequence i~ < i 2 < . . . such that Mij is solvable, (n, [M~jI) = 1 and GUMij is either alternating or of Lie type over a field of characteristic q~ where (qi, n) = 1. D e f i n i t i o n. An element 9 e G is called semisimple if it is semisirnple with respect to some Kegel sequence. D efi n i t i o n. A group G e ~, if and only if G has a series of finite length in which the number of non-abelian simple factors is less than or equal to n and the other factors are locally solvable. If we show that the locally finite simple group G in Theorem 3 has a semisimple element, then it is clear that Theorem 3' implies Theorem 3. By assumption we have a Kegel sequence (G~, N~)~N in which all N~ = 1. By passing to a subsequence we may assume that there is an infinite sequence (it) i 1 < i2 < ... such that for all it: 1) Gi, are alternating groups. 2) G~ are groups of fixed Lie type with unbounded rank parameter over a field K~. of odd characteristic. 3) G~. are groups of fixed Lie type with unbounded rank parameter over a field Ki. of even characteristic. In case (1) any element of order n in G~ is a semisimple element. In case (2) we take an involution as a semisimple element. In case (3) any element of odd order is a semisimple element. We prove the Theorem 3' as a result of the Lemma 4.1. P r o o f o f T h e o r e m 3 ' . Let s be a semisimple element of order n. Assume if possible that such a locally finite simple barely transitive permutation group exists. Then by (2.11) G is not a linear group. Then by [13] Theorem (4.7) CG(s) ~ ~n+2 and involves an infinite simple group. Since G is simple, CG(s ) is a proper subgroup of G. Hence by (2.13) C~(s) is residually finite. But by (4.1) a residually finite group in ~,+ 2 cannot involve an infinite simple group. Hence existence of such a simple group is impossible. Lemma 4.1. Let G be a locally finite group in ~,. I f G is residually finite, then G is almost locally solvable. In particular G does not involve infinite simple groups.

P r o o f. Proof is by induction on n. If n = 0, then G is locally solvable. Assume that G ~ ~, and the assertion is true for n - 1. Let G = G o ~- G1 ~ G2 ~ G3... t:> G k = 1 be a series of G of finite length in which the number of non-abelian simple factors is less than or equal to n and the other factors are locally solvable. Assume that GJGi_ 1 is the first simple factor in the series of G. Then G~_ 1 satisfies the hypothesis of the lemma and the number of non-abelian simple factors is less than or equal to n - 1. Hence by induction assumption Gi-1 is almost locally solvable. If we can show that GI/G ~_ ~ is finite, then it is clear that G is almost locally solvable. Assume if possible that GI/G i_ 1 is an infinite simple group. Let K be the locally solvable subgroup of G~_ 1 such that the index of K in G~_ 1 is m. Hence G~_ ~ does not involve simple groups of order greater than m. Since G~ is residually finite, for any finite subgroup X of Gi there exists a normal subgroup N x of Gi such that IGi : Nxl < ~ and X ~ Nx = 1.

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Let X and N x be as above. Then N x G i - 1 / G i - 1 ~_ Gi/Gi_l.

Simplicity of Gi/G i_1 implies that either N x G i _ l = G i _ 1 or N x G I_1 = G i . If Nx Gi-1 = Gi-1 we have N x < Gi_ 1. Thus Gi/G i_ 1 is a finite group which is impossible by assumption. Hence for any finite non-trivial subgroup X of G~ we have N x G~_ 1 = Gi. N o w we will show that under the assumption that, G~/Gi_ 1 is an infinite simple group the group G~_ 1 involves simple groups of order greater than m and hence we obtain a contradiction. By [12] Theorem 4.4 we may assume that G~/G~_ 1 is a countably infinite locally finite simple group. Then by [12] Theorem 4.5 Gi/Gi_ 1 -- 0

H]/GI-1

j=l

where Hj/Gi-1 is finite and has a maximal normal subgroup N J G i - 1 satisfying (HJG~_ 1) c~ (N]+I/G i_ 1) = 1. It is clear that the orders of finite simple factors are strictly increasing. Therefore there exists k such that [(Hk/Gi_I)/(Nk/Gi_I) [ = l > m . Let x 1, x 2 . . . . . x t be the set of elements of Ilk such that ( x l , x2 . . . . . x t ) Nk/GI- 1 = Hk/Gi- 1. Let X = ( x l, x 2 ..... xt). Then we have X "~ X N x / N x < G i _ I N x / N x = G J N x ~- Gi_l/Gi_l n Nx.

Hence G~_~/G~_~ n Nx contains a subgroup isomorphic to X. But X is finite and has a simple factor group of order n > m. Hence G~_ 1 involves simple groups of order greater than m. But this is impossible. In the following theorems we show that if we put some restrictions on the Kegel sequence of G, then G cannot be isomorphic to a barely transitive group. One can apply [10] Theorem B' to barely transitive groups. But for simplicity we apply the following special ease of it to barely transitive groups. Lemma 4.2. I f G is a non-linear locally finite simple group with a Kegel sequence (Gi, Ni)i~ N and Ni < hypercentre o f Gi, then G has a non-trivial element x say o f order n, the group C~ (x) ~ q~,+2 and C 6 (x) involves a non-linear simple group. In particular G cannot be isomorphic to a barely transitive group. P r o o f. By [10] Theorem B' there exist a non-trivial element x such that CG(x) involves a non-linear simple group. For the conclusion use (4.1). Theorem 4.3. I f G = ~) G i, and G 1 < G 2 < ... where Gi = Gn " " Gik, is finite and central product o f Gu with k fixed and Gu/Z(Gu) is a finite non-abelian simple for all i and j, then G is not isomorphic to a barely transitive group. P r o o f. By using the classification of the finite simple groups we may assume that for a fixed j each {Gi, j / Z ( G i f l : i ~ I} is of the same type, {A, B, C, D, E, F, G} or alternating group. We would like to choose an element s e G such that 1 4; s Z ( G u) is semisimple in the finite non-abelian simple groups Gu/Z(Gu) so we can use our previous argument and say CG,j(s) E ~,+2. As the number of simple factors of each Gi is k, we can conclude that CG,(s) ~ ~k(,+ z) for all i. Hence by Lemma (2.3) of [10] CG(s)~ ~k(n+ 2)" We choose our element s in the following way. By [10] Lemma (2.4) the centralizer of any element of order n in the alternating groups is in ~t,/2j+ 1 where In/2] is the integral Archiv der Mathematik 55

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M. KUZUCUO~LU

ARCH.MATH.

part of n/2. If {GI~/Z(G~ ) : i ~ I} is a set of alternating groups, they do not give any restriction on the choice of the element s. So assume that all the factors GijZ(Gi.i) are finite simple groups of Lie type over the field of characteristic p~j. If {Pij : i ~ I} is a finite set, then one of the primes must occur infinitely often and by passing to a subsequence of {G; : i ~ I} we may assume that for a fixed j and for all i, G~j/Z(Gij) is a finite simple group of Lie type over a field of fixed characteristic. If {p~i : i ~ I} is finite for all j, then we repeat the same process for eachj < k. Hence we get a sequence of{G~ : i ~ I} such that G~jZ(G~j) is a group of Lie type of characteristic p~ for all i and we choose our element s in G~ of order relatively prime to pj wherej = 1, 2 .... k. If {p~j : i ~ I} is infinite for some j and if one of the primes say r occurs infinitely often, then by passing to a subsequence of {G~ : i e 1} if necessary we take the G~ such that G~j is a group of Lie type over a finite field of characteristic r for all i. But if each prime occur finitely many times, then by deleting finitely many terms from the sequence {G~ : i e I} we may exclude the characteristic p for a fixed prime p, and take an element s of order p in G 1. This element will be semisimple in all { G U Z ( G 0 : i ~ I}. If necessary we repeat the same process for eachj and we choose the element s. After choosing such an element s we want to prove that G cannot be isomorphic to a barely transitive group. For if one of the factors G~jZ(G O, i ~ I are alternating groups for all i or in one of the classical families with unbounded rank parameter, then by [10] Lemma (2.4) and [13] (3.24) these groups involve simple groups of arbitrarily large orders, provided that the degree of the alternating group or the rank parameter is chosen suffciently large. Then in this case CG(s) involves an infinite simple group, which implies that, G cannot be isomorphic to a barely transitive group. Finally consider the case that all the direct factors are finite simple groups of Lie type of bounded rank parameter. In this case by passing to a subsequence of {G~: i e I}, if necessary, we may assume that the rank parameter of each {GUZ(G~j ) : i E 1} is fixed and of type ~bj say. Then we have a group of type ~1 . . . . . ~k" We invoke Proposition (3.3) of [8]. Then G = H 1 . . . H g central product of Hi and HJZ(H~) is simple. We have shown in (4.2) that if G has a local system consisting of finite quasisimple groups, then G is not isomorphic to a barely transitive group. In the above case we have that G is a direct product of two proper subgroups. Hence by (2.10) G cannot be isomorphic to a barely transitive group. 5. Closing remarks. The proof of Theorem 3' suggests that one might be able to prove that a proper subgroup of a locally finite barely transitive group cannot involve an infinite simple group, on the grounds that such a proper subgroup must be residually finite. We do not know, if this is true, and the following result of Hartley and Turau must be borne in mind here. We would like to thank B. Hartley and V. Turau for allowing us to include the theorem in this article. Theorem 5.1. I) I f G has a local system orgY-subgroups, then G is homomorphic image of a group such that i) G is residually an W-group i.e. the intersection of the kernels of the homomorphisms of G onto ~{-groups is trivial (note that G contains the restricted direct product of the Gi's). ii) Every finitely generated subgroup of G can be embedded in a direct product of finitely many .~Y-groups. Hence lI) Every locally finite group is an image of a locally and residually finite group. III) Every locally solvable group is an image of a group which is locally and residually sovlable.

Vol. 55, 1990

Barely transitive permutation groups

531

P r o o f. Let G be a group having a local system of subgroups {GI : i ~ I}. This means I is a directed set (a partially ordered set and given i,j ~ I there exists k E I such that i < k and j < k) and i < j implies that G1 < Gi. Let G be any group; {Gi:i E I} the set of finitely generated subgroups. Let (7 = I ] G~, the cartesian product of subgroups G,, and H = {g = (gl)~, : {gl :i ~ I} is a finite set} iel

consists of the elements whose components come from some finite subset of G. It is a subgroup. Now if g(1), . . . , g(n) ~ H

then there exists a partition I = 11 u . . . u I k of I into finitely many subsets, on each of which the elements g (1). . . . . O(") are all constants. Choose ir~l,(1 < r < k). Then each element x ~ (g (1).... , g(")) is constant on each It, so the intersection of the kernels of the projections of (g (1).... , g(")) onto Gil, .... G~ is trivial. Thus (1) each finitely generated subgroup of H is isomorphic to a subgroup of a finite direct product of G~'s. For example if G is locally finite and the Gi are finite, then H is locally finite. Let G={g=(gi)eH: there exists io ~ I such that g~=gio forall i > i o } . The local system property implies that ifjo has the same property, then gio -- g~o. Clearly (~ is a subgroup also. Now define q5 : ~ ~ G as follows. Given g ~ G, choose io as above and put ~b(g) = gioBy the above remark, ~b is well defined. It is a homomorphism, as we see, given g, h ~ ~, by considering an element k o such that gl = gko, hl hko, for all i > k o and noting ~b(g)= gko, ~(h) = hko. Clearly it is surjective. For given 2 e G, choose i0 such that x ~ Oio and define g e t~ by gi = x i f i > io; g1 = 1 otherwise. Then ~b(g) = x. The kernel N of ~b consists of all g e t~ such that there exists io such that g1 = 1 for all i > io. Let g = (gi) E ~ and h(1)..... h (")e N there exist io such that hlj) = 1 for all i > io and 1 < j < n. Put g* = gl if i ~_ i o and g* = 1 if i > i o. Then g* ~ N and ha = h~* where (1 < i < n). Thus g* acts on N by a locally inner automorphism. =

A c k n o w 1e g m e n t. I am grateful to my research supervisor Professor Brian Hartley for his help and encouragement during my study at Manchester. I would like to thank Manchester University and ORS for financial support. References [1] V.V. BELYAEV,Groups of Miller Morena type. Sibirsk Math. Zh. 19, 509-514 (1978). [2] V. V. BELVAEV,Minimal non-FC-groups. All union Symposium on group theory Kiev, 97-108 (1980). [3] V. V. BELYA~V,Locally finite groups with Cernikov Sylow p-subgroups. Algebra and Logic 20, 393-402 (198l). [4] R.W. CARTER, Simple groups of Lie Type. London 1972. [5] P. HALL and C. R. KFLATILAKA,A Property of Locally Finite Groups. J. London Math. Soc. 39, 235 239 (1964). [6] B. HARTLEY, A note on the normalizer condition. Proc. Cambridge Philos. Soc. 74, 11-15 (1973). [7] B. HARTLEY,On the normalizer condition and Barely Transitive Permutation Groups. Algebra and Logic 13, 334-340 (1974). [8] B. HARTLEY, Fixed Points of Automorhpisms of certain locally finite groups and Chevalley Groups. J. London Math. Soc. (2) 37, 421-436 (1988). [9] B. HARTLEY and G. Snua~, Monomorphisms and direct limits of finite groups of Lie type. Quart. J. Math. Oxford (2) 35, 49-71 (1984). [10] B. HARTLEYand M. Kuzucuo~LU, Centralizers of Elements in Locally Finite Simple Groups. To appear. [11] H. HEINEKEN and I.J. MOHA~ED, A Group with Trivial Centre Satisfying the Normalizer Condition. J. Algebra 10, 368-376 (1968). [12] O.H. KEGEL and B. WEHRFRITZ, Locally finite Groups. Amsterdam 1973. [13] M. KLrZUCUOdLU, Barely Transitive Permutation Groups. Thesis University of Manchester 1988. 34*

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M. Kuzucuo~Lu

ARCH. MATH.

[14] A. Ju. OL'SANSKII,An Infinite group with Subgroups of Prime Orders. Math. USSR Izv. 16, 279-289 (1981). [15] D. J. S. RO~II~SON,A Course in the Theory of Groups. Graduate Texts in Math. 80, BerlinHeidelberg-New York 1982. [16] V.P. SUNKOV,On the Minimality Problern for Locally Finite Groups. Algebra and Logic 9, 137-151 (1970). Eingegangen am 26. 4. 1989 Anschrift des Autors: Mahmut Kuzucuo~lu Department of Mathematics Middle East Technical University Ankara 06531 Turkey