ISRAEL JOURNAL OF MATHEMATICS 185 (2011), 293–316 DOI: 10.1007/s11856-011-0111-8

ON THE SUBGROUP INDEX PROBLEM IN FINITE GROUPS BY

M. Costantini and G. Zacher Dipartimento di Matematica Pura ed Applicata, Universit` a di Padova, Via Trieste 63, 35121 Padova, Italy e-mail: [email protected] and [email protected] In memoriam Michio Suzuki

ABSTRACT

We address the subgroup index problem in a given finite subgroup lattice L = (G) which is P -indecomposable and determine out of the structure ˜ invariant for all automorphisms of L the existence in G of a subgroup D of L, with a cyclic complement R in G and where for any pair X ≤ Y of ˜ the index | Y : X | can be computed using only structural subgroups of D properties of L. As a consequence, we show that in such an L all the terms of the Fitting series of G can be determined, as well as an upper bound of the order of G can be computed out of L as long as G has no cyclic Hall direct factor.

1. Introduction For a given group G, the set (G) of all its subgroups, partially ordered by inclusion, is a complete algebraic lattice. A lattice L is called a subgroup lattice if there exists a group G and an isomorphism ψ of L onto (G): then G is determined by L within a projectivity of G. The class Lg of subgroup lattices has been characterized in [Y] (see also [S2 ], Theorem 7.1.17). Lg is properly contained in the class of complete algebraic lattices, as easy examples show. Given L ∈ Lg , i.e., L ∼ = (G), a central problem in the study of groups from a lattice-theoretical point of view is to detect group properties which are encoded Received May 24, 2009 and in revised form November 4, 2009

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in the structure of L: it refers to group properties shared by all groups G for which (G) ∼ = L, i.e., by all elements in the projective closure {G} of the group class {G}. Given ψ : L → (G), let [b/a] be an interval of L such that for x ≤ y in [b/a] the index | y ψ : xψ | can be determined by (L, ψ), i.e., can be computed using only the structural properties of L given by its identification with (G) via ψ. We then say that (L, ψ) determines the indices in [b/a]. With this in mind we propose Definition 1.1: Given L ∈ Lg , the interval [b/a] of L is called a di-interval if (L, ψ) determines the indices in [b/a] for every ψ : L → (G) and the values of the indices are independent of the choice of ψ. Note that in a given L ∈ Lg the interval [b/a] is a di-interval if and only if for a given isomorphism ψ : L → (G) and for x ≤ y in [b/a] the index | y ψ : xψ | can be computed out of (L, ψ) and if for any projectivity ϕ : G → G, ϕ|[bψ /aψ ] is index preserving. An L ∈ Lg is called a di-lattice if b = I, a = 0. For simplicity we shall usually identify an L ∈ Lg with (G) and simply write L = (G). The subgroup index problem associated to L = (G) consists in detecting di-intervals [B/A] in L as well as to indicate lattice procedures in L suited to calculate the indices | Y : X | for all X ≤ Y in [B/A]. Motivated by [DFdGMS], the present paper will address the subgroup index problem for elements of the class LF of finite subgroup lattices, i.e., of finite groups. The paper [SZ] deals with the same kind of problem, but mainly in classes of non-periodic groups. The theory of singular projectivities introduced by M. Suzuki in his 1951 seminal paper [Su1 ] on subgroup lattice questions, and further developed by Schmidt in [S1 ], not only provides basic tools to deal with the subgroup index problem, but also sets bounds to what extent precise answers can be expected in general. The paper is structured as follows. In section 2 we present a series of preliminary results such as lattice characterizations of projectively closed group classes relevant to our investigation, as well as lattice criteria suited to discover di-intervals and to compute in lattice-theoretical terms subgroup indices. Section 3 contains a main structure result; we present for a given L = (G) ∈ LF , with G P -indecomposable (see [S2 ], p. 170), a lattice procedure to derive for

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˜ : R where D ˜ is a subG in a series of steps a split decomposition G = D ˜ group invariant under all automorphisms of L, with [D/1] a di-interval and R a cyclic complement. As an application of the main result, section 4 contains criteria to recognize in lattice-theoretical terms for a given L = (G) ∈ LF the terms Fk (G) of the Fitting series for k ≥ 2 and even for k ≥ 1 if G is P -indecomposable. As another application of the main result, in section 5 for a given L = (G) ∈ LF , where G has no cyclic Hall direct factor, we present a formula to compute a more accurate upper bound for the order of G, improving the one given by Suzuki in [Su1 ], Theorem 1. Apart from some ad hoc conventions, our notation and terminology are standard, and can be found in [R], [Su2 ], [S2 ]. Also, K ≤ G stands for K a d

Dedekind subgroup, M <· G M maximal subgroup, A  ·B A maximal normal subgroup, Gp a Sylow p-subgroup, C a finite chain, G = N : K, K is a complement of the normal subgroup N in G, P (G) the group of autoprojectivities, X P (G) = X σ | σ ∈ P (G), PX (G) = {σ ∈ P (G) | X σ = X}, PX,Y (G) = {σ ∈ P (G) | X σ = X, Y σ = Y }. If X is a group class, LX stands for {(G) | G ∈ X } ⊆ Lg ; the classes X and LX are mutually determined if and only if X coincides with its projective closure X . For the definition of the group classes P∗ , P∗1 , P∗0 , P, P (n, p) and some of their lattice properties we refer the reader to [Su2 ] and [S2 ]. If p is a prime, we denote by Fp the field with p elements. We note that throughout the paper L = (G) means that L is a member of LF , i.e., G is a finite group. 2. Preliminaries Basic to the subgroup index problem is the following Proposition 2.1: Given L = (G), H ≤ G, then [H/1] is a di-interval if and only if for each atom A of [H/1], [A/1] is a di-interval. Proof. Let K be a cyclic subgroup of H; then [K/1] ∼ = ×[Kp /1] implies (∗)

|K : 1| =



p

p

length Kp

p

where p = | Y : X | for 1 ≤ X <· Y ≤ Kp . Given 1 = F ≤ H, let {Ci | i ∈ I} be the set of all the maximal cyclic subgroups of F . By the principle of inclusion

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  and exclusion ([Ha], 2.1.1), | F : 1 | = J (−1)|J |+1 | j∈J Cj |, where the sum is over all non-empty subsets J of I, and using (∗) one concludes. Theorem 1 in [SZ] shows that for any group G (even infinite), (G) is a di-lattice if and only if [g/1] is a di-interval for all g ∈ G. Corollary 2.2: Assume that Q ≤ H ≤ G, with Q a PH (G)-invariant subgroup of G. Then [H/1] is a di-interval if and only if [Q/1] and [H/Q] are di-intervals. Proof. For g ∈ H, ψ : [g, Q/Q] → [g/Q ∧ g], X → X ∧ g is an index preserving lattice isomorphism; since [g/Q ∧ g] and [Q ∧ g/1] are then diintervals, such is [g/1]. Hence [H/1] is a di-interval by Proposition 2.1. We recall that the projective closure P of the class of all finite primary groups P is P ∪ P ([S2 ], Theorem 2.2.6), while P ∪ P∗1 is the class of finite groups with lower semimodular subgroup lattices and with all intervals irreducible ([Su2 ], Proposition 1.5). Thus a lattice description of the class P∗1 will give us one of the group class P1 = P\P∗1 , i.e., of all finite primary groups which are not (abelian) P -groups. For an alternative lattice description of P1 see [S2 ], Theorem 7.4.10. We begin by quoting Lemma 2.3 in [SZ]. Lemma 2.3: Given L = (G), then G is a product of a cyclic normal p-subgroup N by a cyclic q-subgroup K where p, q are different primes, but G is neither cyclic nor a P -group if and only if the following conditions are satisfied: i) (G) is irreducible and contains two maximal chains N , K, not both atoms, such that G = N, K, where N is a P (G)-invariant complement of K in G; ii) if length[N/1] ≥ 2 and X, Y , K1 , M are the subgroups defined by 1 <· K1 ≤ K, 1 <· X <· Y ≤ N and Y <· Y, K1  = M , then M is either cyclic or M is not modular with [M/1] ∼  [D8 /1] (here D8 is the = dihedral group of order 8). Moreover, in this situation, [N/1] is a di-interval and Z(G) is the maximal subgroup C in K such that [N C/1] is reducible. We observe that, in particular, if G satisfies conditions i) and ii) of Lemma 2.3, then the prime p (a certain power of which is the order of N ) is determined by (G): we then say that G satisfies conditions i) and ii) of Lemma 2.3 for p.

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A lattice description for an element G ∈ P∗1 \ P for a fixed prime p is now at hand. We recall ([Su2 ], p. 12) that G is a P∗1 -group if G contains a normal subgroup S such that S is an abelian p-group for a prime p, G/S is cyclic of order q, q a prime, and the automorphisms of S induced by elements of G have the form a → ar for every a ∈ S and r is independent of a, satisfying r ≡ 1 mod p, rq ≡ 1 mod p (this is equivalent to G = SQ, where S is a normal abelian p-subgroup of G, | Q | is a prime dividing p − 1 and Q induces nontrivial power automorphisms on S, [S2 ] p. 390). Suppose G ∈ P∗1 \ P. Then G/Φ(S) is a (non)-abelian P -group, and we say that G ∈ P∗1 \ P for the prime p if G/Φ(S) ∈ P (r, p). Corollary 2.4: Given L = (G), then G ∈ P∗1 \ P for the prime p if and only if: i) G = S, K where S is the unique P (G)-invariant maximal subgroup of G, Φ(S) = 1, S contains chains of length 2, K is an atom and G/Φ(S) ∈ P (r, p); ii) for each chain N in S, if N is an atom then N, K ∈ P (2, p), while if length[N/1] ≥ 2, N, K satisfies the conditions i) and ii) of Lemma 2.3 for p. Proof. Assume G ∈ P∗1 \ P for the prime p. Then G = S : K where S is an abelian p-subgroup and | K | is a prime dividing p − 1. For a σ ∈ P (G) we must have | S σ | = | S | since G is not in P, hence S σ = S and the necessity of the conditions follows. For the sufficiency, if N is an atom of S, since N, K ∈ P (2, p) and N = N, K ∧ S, N is normal in N, K so | N | = p. If length[N/1] ≥ 2 then, by Lemma 2.3, N is a normal Sylow p-subgroup in the metacyclic group N, K with | K | = p. So S is a Sylow p-subgroup on which K acts non-trivially by power automorphisms: in particular, by [H1 ] Hilfssatz 2.5, S is an abelian pgroup. Since S is not elementary abelian, and K induces power automorphisms, G ∈ P∗1 \ P for the prime p. Remark 2.1: Through Corollary 2.4, the subgroup lattice class LP1 is now well described as the class LP∪P∗1 \ LP∗1 . We point out that if L = (G) ∈ LP1 and if L is not a chain, then L is a di-lattice: in fact, by [S2 ], Theorem 2.2.6, if 1 ≤ X < Y ≤ G then |Y : X | = plength[Y /X] , with p given by G/Φ(G) ∈ P (n, p).

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Definition 2.1: For a given L = (G) we set P1,G = {X ≤ G | X ∈ P1 , ⊆ },

PG = {X ≤ G | X ∈ P, ⊆ }

while m(P1,G ) and m(PG ) denote the maximal elements of the p.o.-sets P1,G and PG , respectively. We shall write H ∈ PG ∩ P (n, p) to emphasize the parameters n and p which are uniquely determined by [H/1] ([S2 ], 2.2). Definition 2.2: For an atom A of L = (G), we set HA = {Hi ∈ PG ∩ P (ni , pi ) | A ≤ Hi , i ∈ IA }. If HA = ∅, we set p = min{pi | i ∈ IA } = pi0 . We put As = {A atom in m(P1,G ) | HA = ∅}. Proposition 2.5: Let S be an element of a given L = (G): i) if S is not an atom, then S ∈ m(P1,G ) if and only if S is a Sylow subgroup of G but not an elementary abelian one. ii) if S = A is an atom, then A ∈ m(P1,G ) if and only if the Sylow subgroups of G containing S are elementary abelian. In particular, the atoms A of m(P1,G ) for which HA = ∅ or A not normal in H for any H ∈ HA are Sylow subgroups of G. If S ∈ m(P1,G ) and is not a chain, then [S/1] is a di-interval. Proof. We recall that P1,G is the p.o.-set of all primary subgroups of G which are not abelian P -groups; therefore, its maximal elements are either Sylow subgroups of G which are not P -groups or atoms of G contained in elementary abelian Sylow subgroups. The conclusion follows by Remark 2.1 and [S2 ], Theorem 2.2.6. According to Proposition 2.5, all elements of As are Sylow subgroups of G. By Suzuki’s theorem ([Su2 ], Theorem 4), the lattice (G) is the direct product of irreducible lattices (Gi ), where G = G1 × · · · × Gh is a Hall factorization. Definition 2.3: Set L˜ = {L ∈ LF | L irreducible and L ∈ LP }. Note that if L = (G) ∈ L˜ and ϕ : G → G is a projectivity, then (G) ∈ L˜ and ϕ (and ϕ−1 ) can’t have a singularity of the second kind ([Su2 ], Proposition 2.9). For convenience of the reader, we recall that G is P -indecomposable (as defined in [S2 ], p. 170) if and only if G is the Hall direct product of subgroups ˜ for each i. Gi , with (Gi ) ∈ L

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Remark 2.2: We collect in this remark some useful criteria suitable to detect normal subgroups in L = (G). (a) If K ≤ T ≤ G, the fixed points in [T /1] under PT,K (G) are subgroups of T normalized by K. Assume X is a Hall subgroup of T (resp. of G) and K a complement of X in T (in G). If PT (G) (P (G)) presents no singularities for the primes dividing | K |, then X  T (X  G) if and only if X is PT (G)-invariant (P (G)-invariant). Proof. If X σ = X for every σ ∈ PT (G), then clearly X  T . So assume X  T , and let σ ∈ PT (G). If X σ = X, then | X σ : X σ ∧ X | = | XX σ : X | would divide | T : X | = | K |; so for every prime p dividing | X σ : X σ ∧ X | the projectivity σ −1 would be singular at p, a contradiction. ˜ T ≤ G, S is a cyclic Sylow subgroup of T and NS (b) Assume L = (G) ∈ L, a complement of S in T . Then NS  T if and only if NS is PT,S (G)-invariant. In particular, for T = G we obtain that if NS is a complement of S in G, then NS  G if and only if NS is PS (G)-invariant. Proof. If NSσ = NS for every σ ∈ PT,S (G), then clearly NS  T . So assume NS  T , and let σ ∈ PT,S (G). If length S ≥ 2, then NSσ is quasinormal in T , by [S2 ], Lemma 5.1.3, so that from T = SNSσ it follows that | NSσ | = | NS |, and NSσ = NS . Assume S is an atom. Then, by [S2 ], Theorem 4.2.6 (b), σ is | S |-regular. Then NSσ is normal in T by [Su2 ], Proposition 2.11, and again it follows that NSσ = NS . ˜ let H ∈ PG ∩ P (n, p) and let ϕ : G → G be Lemma 2.6: Given L = (G) ∈ L, a projectivity. 1) If ϕ = σ ∈ P (G), then σ is p-regular. 2) If H  = 1, then ϕ is p-regular. If H  = 1 and ϕ is p-singular, there exists an atom A < H such that p = | A | = | Aϕ | and A has a normal complement in G. Proof. 1) Assume σ has a singularity at p, i.e., there exists an atom A such that p = | A | = | Aσ |. Since σ is of the first kind, if S is a Sylow subgroup with A ≤ S, S has a normal complement N . Therefore H ∧ N = 1, since H ∧ N  H and p does not divide | N |, i.e., H  = 1, so S is elementary abelian, not cyclic and we must have | S σ | = | S |, since S σ = (S σ )p : Aσ leads to the contradiction N ∧ S σ = Aσ = 1.

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2) Assume H  = 1. If ϕ is singular at p, then G = N : Gp and so H ∧N = 1, a contradiction. Assume H  = 1 and ϕ singular at p. Then the conclusion follows from [S2 ], Theorem 4.2.10. ˜ If H ∈ PG ∩ P (n, p), H = Hp : A and | A | Corollary 2.7: Let L = (G) ∈ L. is determined out of L, then [H/1] is a di-interval. Proof. By Lemma 2.6, every σ ∈ PH (G) is not singular at p. Since H is nonabelian, {Aσ | σ ∈ PH (G)} is then the set of all atoms of H of order | A |. Lemma 2.8: Given L = (G), let A be an atom of m(P1,G ). Then: 11 ) if HA = ∅ and A is normal in at least one Hi ∈ HA , then A  Hi0 ∈ P (ni0 , pi0 ), | A | = p = pi0 ; ˜ HA = ∅ and A  Hi for all i ∈ IA , then A is a Sylow subgroup 12 ) if L ∈ L,  of G and | A | pi − 1 for all i; if | A | is known, then the orders of all atoms of Hi are known; 2) if A has at least two PA (G)-invariant complements, then [A/1] is a di-interval. Proof. 11 ) If A  Hj ∈ P (nj , pj ), then | A | = pj and since | A | ≤ pi for all i ∈ IA , we get | A | = p = pi0 . 12 ) For all i ∈ IA , Hi = 1 and since the Sylow subgroups S containing A are elementary abelian, A would be normal in at least one Hi if A = S. The conclusion follows from Corollary 2.7. 2) Let K1 , K2 be distinct PA (G)-invariant complements of A and ϕ : G → G a projectivity. Then G/K1 ∧ K2 is in P (2, p) and abelian, so | A | = p and ϕ|G/K1 ∧ K2 is index preserving. ˜ let S be a chain in m(P1,G ) and assume Lemma 2.9: Given L = (G) ∈ L, that S ∈ As if S is an atom. If S has no PS (G)-invariant complement, then [S/1] is a di-interval. Proof. By Proposition 2.5, S is a Sylow subgroup of G. Suppose S has a normal complement NS . Then, by Remark 2.2 (b), NS is PS (G)-invariant, a contradiction. Therefore S has no normal complement, hence we must have N (S) > C(S). Choose an M ≤ N (S) such that C(S) <· M and take a chain C in M such that C ≤ C(S). If one sets T = S, C, then S is a normal Sylow

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subgroup of T which is metacyclic, not cyclic, and not a P -group since S ∈ As if S is an atom. Thus, by Lemma 2.3, [S/1] is a di-interval. ˜ let A be an atom of L. Then A is Lemma 2.10: Given an L = (G) ∈ L, quasinormal in G if and only if A ≤ G. d

Proof. If A ≤ G but not quasinormal, then G = AG × T , a Hall factorization d

with AG a P -group (see [S2 ], 5.1.14), a contradiction. Lemma 2.11: Given L = (G), let T = C, H be a subgroup of G with (T ) ∈ ˜ C a chain and H ∈ PG ∩ P (n, p) a PT,C (G)-invariant subgroup. Set L, V = {A | 1 <· A < H, A = A, C ∗ ∧H, C ∗ a chain and a supplement of H in T }. Then an atom A of (H) has order different from p if and only if A ∈ V and A < T. d

Proof. Assume A ∈ V. By Lemma 2.10, A is normal in T if and only if A ≤ T . d

Thus if A ∈ V and A < T , then | A | = p. d

Conversely, let A be an atom of H with | A | = p: we show that A is in V. We have H = Hp : A. From T = HNT (A) and T /H ∼ = NT (A)/A it follows that NT (A) = AC ∗ for a certain chain C ∗ . Then T = HAC ∗ = HC ∗ and AC ∗ ∧ H = NH (A) = A, hence A ∈ V. As already observed, if A ≤ T , then A  H, a contradiction.

d

Remark 2.3: Assume ϕ : G → G is a projectivity and H ∈ PG . If we can embed H in a subgroup T of G satisfying the conditions of Lemma 2.11, then we can decide, out of (G), whether H is abelian or not. In particular, H is abelian if and only if H ϕ is abelian. 3. Structure theorems ˜ we are going to derive a first structure theorem for Given L = (G) ∈ L, G in case P (G) contains singular autoprojectivities, the content of which is essentially that of Theorem 4.2.5 in [S2 ]. ˜ then there are autoprojectivities singular Lemma 3.1: Given L = (G) ∈ L, on a Sylow subgroup S of G if and only if S is a chain in m(P1,G ), with a (unique) PS (G)-invariant complement NS , and for a certain σ ∈ P (G) one has

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[G/NS ∧ NSσ ] ∼ = [S/1] × [S/1]. Assume this is the case. If one sets MS =  σ N , then MS is a P (G)-invariant Hall subgroup with a cyclic, not σ∈P (G) S primary, complement X = X1 × · · · × Xr , where 2 ≤ r = | P (G) : PSMS (G) |, and X1 , . . . , Xr are chains of the same length. In particular, MS = MS τ for any τ ∈ P (G), P (G) acts transitively on the set of maximal chains of [G/MS ] and {X τ | τ ∈ P (G)} is the set of all complements of MS . Proof. Suppose S is a Sylow subgroup of G and there is σ ∈ P (G) such that | S | = | S σ |. Then by [S2 ], Theorem 2.2.6, and Lemma 2.6, S is a chain, and therefore an element of m(P1,G ). By [S2 ], Theorem 4.2.6, S has a (unique) normal complement NS which is PS (G)-invariant by Remark 2.2 (b). Since | S | = | S σ |, we get G/(NS ∧ NSσ ) ∼ = S × S σ , i.e., [G/(NS ∧ NSσ )] ∼ = [S/1] × [S/1] σ (note that since NS is PS (G)-invariant, then NS is PS σ (G)-invariant, so that NSσ is normal in G and it is the unique PS σ (G)-invariant complement of S σ ). Conversely, assume that for a chain S ∈ m(P1,G ) with a PS (G)-invariant complement NS there is a σ ∈ P (G) such that [G/(NS ∧ NSσ )] ∼ = [S/1] × [S/1]. σ We show that | S | = | S | and that S is a Sylow subgroup. Let τ ∈ P (G). Since NS is PS (G)-invariant, it follows that NSτ is PS τ (G)-invariant and therefore normal in G. In particular, both NS and NSσ are normal in G, G/(NS ∧ NSσ ) is isomorphic to a subgroup of S × S σ . But [G/(NS ∧ NSσ )] ∼ = [S/1] × [S/1] then σ implies that | S | = | S |. If length S ≥ 2, then S is a Sylow subgroup. Assume S is an atom. If S is not a Sylow subgroup of G, let T be a Sylow subgroup containing S. By Lemma 2.6, σ is not singular on T , a contradiction, since | S | = | S σ |. Hence S is a Sylow subgroup of G. We observe that if τ ∈ P (G), it follows that NSτ is the unique PS τ (G)-invariant complement of the Sylow subgroup S τ of G: hence NSτ = NS τ . The rest of the assertions are now clear. ˜ set Given L = (G) ∈ L, S = {S | S a chain in m(P1,G ) such that the conditions of Lemma 3.1 hold}. Therefore, there are singular autoprojectivities if and only if S = ∅. It is now straightforward to derive the following structure result: ˜ assume S = ∅. Then K ˜ = Proposition 3.2: Given L = (G) ∈ L, S∈S MS is a P (G)-invariant Hall subgroup of G with a cyclic complement Z, (1)

˜ : Z, G=K

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Z is the product of cyclic Hall subgroups Z = Z1 × · · · × Zt , t ≥ 1, where Zi = Xi,1 × · · · × Xi,ri , 2 ≤ ri = | P (G) : PXi,1 MXi,1 (G) | and Xi,h , Xi,k are ˜ is index chains of the same length for h, k = 1, . . . , ri . For any σ ∈ P (G), σ|K preserving. Definition 3.1: For a given L = (G) ∈ L˜ we shall call the decomposition ˜ : Z given above when S = ∅, a (1)-decomposition of G. If S = ∅ we put G=K ˜ K = G, Z = 1. We come to a corollary already known to Suzuki (see [Su2 ], p. 51) ˜ let σ ∈ P (G) and R  ·A ≤ G. Then Corollary 3.3: Given L = (G) ∈ L, σ σ R  ·A . In particular, σ preserves the sets of subnormal and of quasinormal subgroups of G. The Fitting subgroup F (G) is P (G)-invariant and the same ˜ holds for the Sylow subgroups of F (G) ∧ K. Proof. If A/R is simple non-abelian, then Rσ  ·Aσ by [Su2 ], Theorem 14. So ˜ ∧ Aσ ˜ ∧A = K ˜ ∧ R. Then, since K ˜ σ ∧ Aσ = K assume | A : R | = p. Suppose K σ σ σ  σ σ σ ˜ ∧ A ) is cyclic, we get (A ) ≤ K ˜ ∧A = K ˜ ∧ R ≤ R , so that and A /(K  σ σ ˜ ˜ ˜ |, so R  A . Otherwise we get p = | K ∧ A : K ∧ R |, and in particular p | K σ σ that σ is p-regular. Then R  A by [Su2 ], Proposition 2.11. Since a subnormal subgroup is quasinormal if and only if it is a Dedekind subgroup, the conclusion follows. ˜ We observed that if K ≤ T ≤ G, the fixed Remark 3.1: Let L = (G) ∈ L. points in [T /1] under PT,K (G) are subgroups of T normalized by K. Assume X is a Hall subgroup of T (resp. of G) and K a complement of X in T (in G). Then X  T (X  G) if and only if X is PT,K (G)-invariant (PK (G)-invariant). Proof. If X σ = X for every σ ∈ PT,K (G), then clearly X  T . So assume X  T , and let σ ∈ PT,K (G). By Corollary 3.3, X σ is quasinormal in T . Then X σ = X, since X σ is a complement of K in T . Remark 3.2: Given L = (G) ∈ L˜ and a projectivity ϕ : G → G, it is clear ˜ : Z is a (1)-decomposition for G, then G = K ˜ ϕ : Z ϕ is a that if G = K (1)-decomposition for G. We also note that an element σ in P (G) is index preserving if and only if ˜ = 1. In particular, the group I(G) of index preserving autoprojectiviσ|[G/K] ties of G is normal in P (G).

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˜ We proceed now to a further lattice decomposition for an L = (G) ∈ L, ˜ : Z, by considering the set Y of starting from a (1)-decomposition G = K chains S of m(P1,K˜ ), with S ∈ As if S is an atom, having a normal (hence P (G)-invariant by Remark 2.2 and Proposition 3.2) complement NS in G: set   ˜∧ Z˜ = K NS . S∈Y

Then we get for G the decomposition (2)

G = Z˜ : (Z1 × Z),

where Z˜ is a P (G)-invariant Hall subgroup and where Z1 × Z is a cyclic com˜ = Z˜ : Z1 . plement, with K Definition 3.2: For a given L = (G) ∈ L˜ we shall call the decomposition G = Z˜ : (Z1 × Z) given above, a (2)-decomposition of G. ˜ let G = Z˜ : (Z1 × Z) be a (2)Proposition 3.4: Given L = (G) ∈ L, decomposition of G. If S is a non-atomic element of m(P1,Z˜ ) or an element of ˜ As ∩ [Z/1], then [S/1] is a di-interval. Proof. If S is not a chain, the claim follows from Remark 2.1. If S is a chain, we conclude by Lemma 2.9, since S has no PS (G)-invariant complement, being ˜ S ≤ Z. Remark 3.3: Given L = (G) ∈ L˜ and a projectivity ϕ : G → G, if G = Z˜ : (Z1 × Z) is a (2)-decomposition for G, then G = Z˜ ϕ : (Z1ϕ × Z ϕ ) is a (2)-decomposition for G. ˜ so there exists an atom A ≤ Z˜ such Let ϕ have a singularity at p on Z; ϕ that p = | A | = | A |, and by a (2)-decomposition, if A ≤ Z˜p , then either Z˜p is a chain, so that Z˜p = A, HA = ∅ and A has a normal complement MA which is P (G)-invariant. Otherwise, Z˜p is a P -group and, in particular, HA = ∅: in this case we are in the situation described in [S2 ], Theorem 4.2.11 (b) and again A has a unique normal complement MA which is P (G)-invariant. Moreover, Op (G) is actually a complement of A in Z˜p , and A is not subnormal in G, since (G) is irreducible (if N is the normal complement of Z˜p in G, then MA = N × Op (G) and Z˜p = A × Op (G)). For the index problem, we are interested in discovering atoms of Z˜ which give rise to di-intervals.

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˜ let G = Z˜ : (Z1 × Z) be a (2)-decomposition Lemma 3.5: Given L = (G) ∈ L, of G. 1) If A is an atom of a non-atomic element S of m(P1,Z˜ ), or A is an atom in ˜ As ∩ [Z/1], then [A/1] is a di-interval. 2) If A is an atom of m(P1,Z˜ ) and A has no P (G)-invariant complement in G, then [A/1] is a di-interval. Proof. 1) This follows from Proposition 3.4. 2) If HA = ∅, then we are done by 1). So assume HA = ∅ and let S be a Sylow subgroup of Z˜ containing A: then S is elementary abelian. Suppose S = A. By Remark 3.1, S has no normal complement and, as in the proof of Lemma 2.9, we can choose an M ≤ N (S) such that C(S) <· M and take a chain C in M such that C ≤ C(S). If one sets T = S, C, then T is metacyclic, non-cyclic. If T is not a P -group, then, by Lemma 2.3, [A/1] is a di-interval. Otherwise T is a P -group, and S is normal in T . Then by Lemma 2.8 11 ), | A | is determined and, if ϕ : G → G is a projectivity, | Aϕ | = | A | by Remark 3.3 since A has no P (G)-invariant complement. So finally assume A = S. Then again by Lemma 2.8 11 ), | A | is determined and | Aϕ | = | A | by Remark 3.3. We note that by Lemma 3.5, if A is an atom in m(P1,Z˜ ), then one can determine the order of A either if A lies in As , or if A has no P (G)-invariant complement in G, and no singularity takes place on A. In the first case we know that A is a Sylow subgroup of G. ˜ let G = Z˜ : (Z1 × Z) be a (2)Corollary 3.6: Given L = (G) ∈ L, ˜ decomposition of G. Then [F (Z)/1] is a di-interval. ˜ By Lemma 3.5 1), we may assume that A is an Proof. Take an atom A of F (Z). atom in m(P1,Z˜ ) with HA = ∅; then A is normal in any H ∈ HA , hence again | A | is known by Lemma 2.8 and | Aϕ | = | A | for every projectivity ϕ : G → G by Remark 3.3. Lemma 3.5 and Lemma 2.8 2) suggest the introduction of the following crucial ˜ set of atoms of L = (G) ∈ L: (3) A = {A atom of m(P1,Z˜ ) with a unique P (G)-invariant complement MA in G}. In particular, if A ∈ A, then HA = ∅. According to Remark 3.3, the atoms in A are the only atomic candidates of Z˜ for the existence of a projectivity

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ϕ : G → G such that | Aϕ | = | A |. We come now to the main result of this section. ˜ let G = Z˜ : (Z1 ×Z) be a (2)-decomposition Theorem 3.7: Given L = (G) ∈ L, of G and let A be the set of atoms of m(P1,Z˜ ) defined in (3). Set I =  Z˜ ∧ ( A∈A MA ); then I is a P (G)-invariant subgroup with G ≤ I and there ˜ of G such that I ≤ D ˜ ≤ Z, ˜ F (Z) ˜ ≤ D, ˜ exists a P (G)-invariant subgroup D ˜ contains every chain of length at least 2 of Z, ˜ Z˜ = D ˜ : Z2 , Z2 is a cyclic D ˜ ˜ square-free group, G = D : R, [D/1] is a di-interval, R is a cyclic complement isomorphic to Z2 × Z1 × Z. Proof. For A ∈ A, Z˜ ∧ MA ≥ G and Z˜ ∧ MA is P (G)-invariant: hence I ≥ G , ˜ is abelian, with elementary abelian Sylow I is P (G)-invariant. Moreover Z/I subgroups. Let ϕ : G → G be a projectivity: then I ϕ is the corresponding subgroup in G, since {Aϕ | A ∈ A} is the set of atoms in m(P1,Z˜ ϕ ) with a 

unique P (G)-invariant complement (MAϕ ) in G. Hence G ≤ I ϕ . ˜ = D/I × F/I, the Hall factorization where F/I is the maximal Write Z/I ˜ cyclic (square-free) Hall subgroup of Z/I. Then D and F are P (G)-invariant ˜ and Z/D is cyclic square-free. Let A be an atom of I. Then, by Lemma 3.5, [A/1] is a di-interval. By Proposition 2.1, [I/1] is a di-interval, and so is [D/1]  by Corollary 2.2, since G ≤ I and G ≤ I ϕ imply ϕ|D is index preserving. By ˜ is P (G)-invariant and [F (Z)/1] ˜ Corollaries 3.3 and 3.6, F (Z) is a di-interval, ˜ so that, if we put D1 = D F (Z), then D1 is P (G)-invariant, with [D1 /1] a ˜ di-interval. Since Z/D is cyclic square-free, if C is a chain of Z˜ of length ≥ 2, then C ∧ D1 = 1, so [C/1] is a di-interval. Let C be the set of all chains of Z˜ ˜ is a P (G)-invariant ˜ = D1 , C | C ∈ C. Then D of length at least 2, and let D ˜ subgroup, [D/1] is a di-interval by Corollary 2.2. ˜ is cyclic, since it is abelian and G/K, ˜ K/ ˜ Z˜ and Z/ ˜ D ˜ are Note that G/D ˜ cyclic of pairwise coprime order. Let y ∈ G be such that G = Dy. We write ˜ and Z˜ y = y1 · · · yk , where yi  is a Sylow subgroup of y for every i. Since K are normal Hall subgroups of G, up to reordering {y1 , . . . , yk }, we may assume ˜ = y1 · · · yh  and y ∧ Z˜ = y1 · · · yh  ∧ Z˜ = y1 · · · ys  so that y ∧ K ˜ : yh+1 · · · yk , G=K

˜ = Z˜ : ys+1 · · · yh , K

G = Z˜ : ys+1 · · · yk 

with yh+1 · · · yk  conjugate to Z and ys+1 · · · yh  conjugate to Z1 . We may put Z = yh+1 · · · yk  and Z1 = ys+1 · · · yh .

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˜ ˜ = Dy ˜ 1 · · · ys , and D ˜ contains all the chains of Z˜ of Now Z˜ = D(y ∧ Z) ˜ is a Hall subgroup of y1 · · · ys . length at least 2. It follows that y1 · · · ys  ∧ D ˜ If we assume y1 · · · ys  ∧ D = y1 · · · yt , then ˜ : yt+1 · · · ys , Z˜ = D

˜ =D ˜ : yt+1 · · · yh , K

˜ : yt+1 · · · yk . G=D

At this point we put Z2=yt+1 · · · ys , R=yt+1 · · · yk , so that R =Z2 ×Z1×Z. ˜ is lattice defined, so that D ˜ is We shall see in the next section that F (Z) lattice defined too. ˜ contains every Sylow subgroup of G which is neither Remark 3.4: Note that D ˜ in cyclic nor elementary abelian. Moreover, there exists a complement R of D ˜ : Z, G = Z˜ : (Z1 × Z), G of the form R = Z2 × Z1 × Z such that G = K ˜ ˜ Z = D : Z2 . Corollary 3.8: Given L = (G) ∈ LF , with G P -indecomposable, there ˜ such that G = D ˜ : R, R cyclic and [D/1] ˜ exists a P (G)-invariant subgroup D a di-interval. Proof. Let G = G1 × · · · × Gt , t ≥ 1 be the Hall factorization of G with ˜ i : Ri . If one irreducible Gi ’s. Applying Theorem 3.7 to each Gi , we get Gi = D ˜ ˜ ˜ ˜ sets D = D1 × · · · × Dt , R = R1 × · · · × Rt , then G = D : R as required.

4. Fitting subgroups Given L = (G), if A, K are elements of L, we set C,K (A) = x ≤ K | [A, x/1] ∼ = [A/1] × [x/1] and simply C (A) for K = G. If A is a Hall subgroup of G, then A, C (A) = A×C (A) since N (A) = A : X implies C (A) = CX (A); moreover, C(A) = Z(A)× C (A) so that, in particular, C(A) = A × C (A) if and only if Z(A) = A and C(A) = C (A) if and only if Z(A) = 1. We recall that a non-trivial group K is called quasisimple ([KS], p. 127) if K is perfect and K/Z(K) is simple. Lemma 4.1: Given L = (K), K is quasisimple if and only if no maximal subgroup of K is a Dedekind subgroup and K contains a (Dedekind) subgroup

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Z such that [K/Z] contains no proper non-trivial Dedekind element and for any set X of chain generators of Z we have C (C) = K for every C ∈ X . Proof. We recall that K is perfect if and only if no maximal subgroup of K is a Dedekind subgroup ([S2 ], Theorem 5.3.3). Assume now Z < K, with no proper non-trivial Dedekind element in [K/Z] and with a set X of chain generators of Z, such that C (C) = K for any C ∈ X . Then Z ≤ Z(K), and it follows that Z = Z(K) with K/Z is simple by [S2 ], Theorem 5.3.1. Conversely, assume that K is quasisimple, and let Z = Z(G). Then there are no proper non-trivial Dedekind elements in [K/Z]. Let X be any set of chain generators of Z, and let C ∈ X . Since K is quasisimple, there exists a chain Y of K such that [C, Y /1] ∼ = [C/1] × [Y /1] and K = Y x | x ∈ K. Hence C (C) = K and we are done.

Lemma 4.1 gives a lattice-theoretic characterization of the class of quasisimple groups. In particular this class is projectively closed. Lemma 4.2: Given L = (G), let K be a quasisimple subgroup of G. Then K is subnormal in G if and only if K ≤ K P (G) . d

Proof. If K is subnormal then, for σ ∈ P (G), K σ is subnormal by [S2 ], Corollary 5.4.14. So by [KS], 6.5.2, [K, K σ ] = 1 if K σ = K, and therefore K  K P (G) . Since K is perfect, we have K  K P (G) if and only if K ≤ K P (G) ([S2 ] Theorem d

5.1.7), and the converse follows. As a consequence of Lemmas 4.1 and 4.2, given L = (G), the join E(G) of all quasisimple subnormal subgroups of G is a lattice defined subgroup of G; in particular, it is P (G)-invariant and, if ϕ : G → G is a projectivity, then E(G) = E(G)ϕ . We also observe that since E(G) is perfect, then by Theorem 3.7, [E(G)/1] is a di-lattice, and therefore a di-interval in (G). In particular, for any ϕ : G → G, ϕ|E(G) is index preserving. On the other hand, the Fitting subgroup F (G) is not in general lattice defined, ˜ we are going as the class of P -groups shows. However, in case L = (G) ∈ L, ∗ to show that F (G), hence also F (G) = F (G)E(G), are lattice defined, so that they are P (G)-invariant and for any projectivity ϕ : G → G one has F (G)ϕ = F (G) and F ∗ (G)ϕ = F ∗ (G). We start from a (2)-decomposition

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G = Z˜ : (Z1 × Z) of G, which gives rise to (4)

˜ × (F (G) ∧ (Z1 × Z)) F (G) = F (Z)

˜ where F (G) ∧ (Z1 × Z) = C (Z).

By Corollary 3.3, F (G) is a P (G)-invariant subgroup of G and so, by Proposition ˜ Moreover, C (Z) ˜ is P (G)-invariant 3.2, such are all the Sylow factors of F (Z). ˜ and [F (Z)/1] is a di-interval by Corollary 3.6. ˜ let G = Z˜ : (Z1 × Z) be a (2)-decomposition Lemma 4.3: Given L = (G) ∈ L, of G. Then: ˜ if and only if C P (G) is a primary group; 1) a chain C of Z˜ is in F (Z) 2) a chain C of G with C ∧ Z˜ = 1 is in F (G) if and only if C P (G) is cyclic. ˜ Then C ≤ Op (F (Z)) ˜ for a certain prime Proof. 1) Assume C is a chain of F (Z). P (G) ˜ ˜ p. Since Op (F (Z)) is P (G)-invariant, we get C ≤ Op (F (Z)). Conversely, if ˜ if C P (G) is a primary group, then clearly C ≤ F (G), so that C is a chain of Z, ˜ C ≤ F (Z). 2) Assume C is a chain of G with C ∧ Z˜ = 1. If C ≤ F (G), then, by ˜ which is cyclic and P (G)-invariant. Hence C P (G) is cyclic. (4), C ≤ C (Z) Conversely, if C P (G) is cyclic, then C lies in F (G). We show that Lemma 4.3 gives a lattice characterization of F (G): we just need to prove that we can decide whether C P (G) is a primary group or not in lattice-theoretical terms. The situation is clear if (C P (G) ) belongs to the class LP1 , which is the case when length C ≥ 2. The problem arises when C P (G) ∈ PG . In what follows we propose a solution: (i) if H = C P (G) ∈ PG , then it is a primary group if H is a proper subgroup of an element R ∈ PG ∩ P (n, p), since then H is a p-group, being H  R; (ii) we observe that if for each chain C ∗ not contained in H we have [C ∗ , H/1] ˜ reducible, then we would have L reducible, a contradiction since L ∈ L; ∗ ∗ (iii) we are left with the existence of a chain C such that T = C , H has all the properties of the group T as defined in Lemma 2.11. But then, by Remark 2.3, we can decide in lattice-theoretical terms whether H is abelian or not. We have therefore proved that Lemma 4.3 gives us a lattice characterization of F (G), which is therefore invariant under projectivities. Since both F (G) and E(G) are lattice defined, it follows that the same holds for F ∗ (G). Proposition 4.4: Given L = (G), with G P -indecomposable, then F (G) and ˜ E(G) are lattice defined subgroups and [F (Z)/1] is a di-interval.

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Proof. Let G = G1 × · · · × Gt be the Hall factorization of G, with (Gi ) irreducible for each i = 1, . . . , t. Then F (G) = F (G1 ) × · · · × F (Gt ) and E(G) = E(G1 ) × · · · × E(Gt ), and we conclude by Lemma 4.3, (4) and Theorem 3.7. Our aim is to show that for any (finite) group G the terms Fn (G) of the Fitting series of G are lattice defined for every n ≥ 2. In particular, if G is P -indecomposable, Fn (G) is lattice defined for every n ≥ 1. Lemma 4.5: Given L = (G), then F2 (G) is lattice defined. Proof. Let G = G1 ×· · ·×Gt be the Hall factorization of G with (Gi ) irreducible for every i = 1 . . . , t. Then F2 (G) = F2 (G1 ) × · · · × F2 (Gt ). If Gi is a P -group, ˜ then F2 (Gi ) = Gi , hence we may assume G ∈ L. Set N = F (G), which is lattice defined by Proposition 4.4. If N = 1 we get F2 (G) = 1, and we are done. So assume N = 1. We have F2 (G)/N = F (G/N ), so that, if G/N is P -indecomposable, then F (G/N ) is lattice defined, and we are done. So assume ()

G/N = H1 /N × · · · × Hs /N × X/N

is the Hall direct decomposition of G/N with X/N P -indecomposable and Hi /N ∈ P (ni , pi ) for i = 1, . . . , s. In particular, if σ ∈ P (G), then N σ = N , and Hiσ = Hi , X σ = X for every i = 1, . . . , s by [S2 ], Lemma 4.2.4. From () we get F (Hi ) = F (X) = N and F2 (G) = F2 (H1 ) · · · F2 (Hs )F2 (X). Moreover, F (X/N ) (i.e., F2 (X)) is lattice defined, and our aim is to show that we can determine F (Hi /N ), i.e., F2 (Hi ), out of (G) for each i. ˜ × R, Let G = Z˜ : (Z1 × Z) be a (2)-decomposition of G. Then N = F (Z) ˜ = N ∧ (Z1 × Z), so G/N = ZN/N ˜ where R = C (Z) : (Z1 × Z)N/N , where ˜ ZN/N is a Hall normal subgroup. We get ˜ Hi /N = (Hi ∧ ZN/N ) : (Hi ∧ (Z1 × Z)N/N ) ˜ where Hi ∧ ZN/N is a non-trivial normal Hall subgroup of Hi /N , since Hi ∧ (Z1 × Z)N/N is cyclic. If Hi ∧ (Z1 × Z)N/N is non-trivial, then Hi /N is ˜ non-abelian, F (Hi /N ) = Hi ∧ ZN/N which is lattice defined, and we are done. ˜ ˜ = Z˜ × R. We observe that So assume Hi /N = Hi ∧ ZN/N , i.e., Hi ≤ ZN this implies that every autoprojectivity of G is index preserving on Hi /N , since

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∼ ˜ Z˜ ∧ N and every autoprojectivity of G is index preserving ˜ Hi /N ≤ ZN/N = Z/ on Z˜ (if σ ∈ P (G) and σ|Hi /N is singular at a prime q, then σ is singular at ˜ a q; but q is a divisor of | Hi /N |, hence a divisor of | Z˜ | and σ is regular on Z, contradiction). We claim that to conclude it is enough to show that we can decide out of (G) whether Hi /N is abelian or not. In fact, if it is abelian, then F (Hi /N ) = Hi /N . Otherwise, let Mi /N be the Sylow pi -subgroup of Hi /N . Then Mi is P (G)invariant since every autoprojectivity of G is index preserving on Hi /N , and it is the unique maximal P (G)-invariant subgroup of Hi containing N , for if Ni is another such subgroup, then Ni /N would be a normal maximal subgroup of Hi /N different from Mi /N , so that Hi /N would be abelian, a contradiction. Then F (Hi /N ) = Mi /N is lattice defined. In the following we prove that we can decide out of (G) whether Hi /N is abelian or not. For convenience, we put Hi /N = H/N ∈ P (m, p). We know ˜ × R, Z, ˜ F (Z), ˜ R and the Sylow subgroups of F (Z), ˜ with the that N = F (Z) ˜ corresponding orders, are lattice defined. Since N ≤ H ≤ Z × R, it follows that ˜ (which may be trivial) is lattice defined. p divides | Z˜ |, so that Op (G) = Op (Z) Let us write N = Op (G) × V . In particular, V is lattice defined too. Assume there is a complement T of V in H. If H/N is abelian, then T is a p-group, otherwise T has order pn−1 q, where pn−1 = pm−1 | Op (G) |, and q divides p − 1. Suppose H/N is abelian. Then V is a normal Hall subgroup of H, so that it has a complement T . Therefore (∴) if V has no complement in H, then H/N is not abelian. The next lemmas deal with the situation when V has a complement T in H. Lemma 4.6: Assume V has a complement T in H, with T not a P -group. Then H/N is abelian if and only if T ∈ P1,G (i.e., T ∈ m(P1,G )). Proof. If T ∈ P1,G , then T is a p-group (since we know that p divides | T |), and H/N is abelian. Assume T ∈ P1,G and suppose H/N is abelian. Then T is a p-group, and since T ∈ P1,G , T is elementary abelian. This is a contradiction, since T is not a P -group.

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We are left to deal with the case when V has a complement T in H, with T a P -group. We recall that if V ∧ Z˜ is non-trivial, the factorization V ∧ Z˜ = P1 × · · · × Pr , with Pi the Sylow pi -subgroup of V ∧ Z˜ for i = 1, . . . , r, is lattice defined. Lemma 4.7: Assume V has a complement T in H, with T a P -group. Let V ∧ Z˜ = P1 × · · · × Pr be the factorization of V ∧ Z˜ with Pi the Sylow pi subgroup of V ∧ Z˜ for i = 1, . . . , r, r ≥ 0. Then H/N is abelian if and only if r ≥ 1 and there exists (distinct) maximal subgroups M1 . . . , Ms of T and ˜ s ≥ 2, such that subgroups X1 , . . . Xs of V ∧ Z, (i) Xi , T  ∧ V = Xi for each i = 1, . . . , s; (ii) for each i = 1, . . . , s there exist j ∈ {1, . . . , r} such that Xi ∈ [Pj /Φ(Pj )], Xi = Φ(Pj ), [Xi Mi /Φ(Pj )] ∼ = [Xi /Φ(Pj )] × [Mi /1], [Xi T /Φ(Pj )] irreducible. Proof. Assume all the conditions are satisfied. We prove that H/N is abelian. Suppose for a contradiction that H/N is non-abelian. It follows that T is nonabelian, of order pn−1 q. By (i), Xi is T -invariant. In particular, since T acts on the vector space Pj /Φ(Pj ) over Fpj , it leaves invariant the non-zero subspace Xi /Φ(Pj ). The condition [Xi Mi /Φ(Pj )] ∼ = [Xi /Φ(Pj )] × [Mi /1] means that Mi acts trivially on Xi /Φ(Pj ) and that (| Mi |, | Pj |) = 1. We claim that | Mi | = pn−1 . Otherwise, | Mi | = pn−2 q and Mi is not normal in T . But from Mi ≤ CT (Xi /Φ(Pj )) it follows that CT (Xi /Φ(Pj )) = T since CT (Xi /Φ(Pj )) is normal in T . Hence Xi , T /Φ(Pj ) ∼ = Xi /Φ(Pj ) × T , and this is a Hall decomposition, since (| Mi |, | Pj |) = 1 and p = pj for j = 1, . . . , r. But this is a contradiction, since [Xi T /Φ(Pj )] is irreducible. We have shown that | Mi | = pn−1 for every i = 1, . . . , s. Since T is non-abelian, it has a unique subgroup of order pn−1 , and this is a contradiction, since s ≥ 2. Therefore G/N is abelian. Conversely, assume H/N is abelian: then T is an elementary abelian p-group. ˜ = {1} We claim that the action of T on V is non-trivial (in particular, V ∧F (Z) and r ≥ 1, since R ≤ CG (T )). Suppose for a contradiction that H = V × T . Then F (H) = H. But F (H) = N < H, a contradiction. ˜ on which T acts nonLet P1 , . . . , Ph be the Sylow subgroups of V ∧ F (Z) trivially. For each i = 1, . . . , h we consider Wi = Pi /Φ(Pi ), a vector space over Fpi , and decompose Wi into the direct sum of simple Fpi T -submodules:

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Wi = Wi,1 ⊕ · · · ⊕ Wi,ni . Let ρi,k : T → GL(Wi,k ) be the corresponding representation, and let Mi,k be the kernel of ρi,k . Then either Mi,k is maximal in T , or Mi,k = T , since if W is a non-trivial simple Fpi T -module, then T /CT (W ) is cyclic. We collect in the set Xi all the maximal Mi,k ’s, and we put  Ni = M∈Xi M . Then Ni coincides with CT (Pi ), since (| T |, | Pi |) = 1. Since N = F (H) = V ×Op (G), it follows that CT (V ) = Op (G), so that N1 ∧ · · · ∧ Nh = Op (G). From T /Op (G) ∼ = H/N , it follows that Op (G) is not maximal in T . Therefore in the union X = X1 ∪ · · · ∪ Xh there are at least 2 distinct maximal subgroups. Up to renaming the maximal subgroups in X , we conclude. This concludes the proof of Lemma 4.5. Theorem 4.8: Let G be a finite group. Then Fn (G) is lattice defined for every n ≥ 2. Proof. Let G = G1 ×· · ·×Gt be the Hall factorization of G with (Gi ) irreducible for every i = 1 . . . , t. Then Fk (G) = Fk (G1 ) × · · · × Fk (Gt ) for every k. If Gi ˜ is a P -group, then Fn (Gi ) = Gi for every n ≥ 2, hence we may assume G ∈ L. We make induction on n ≥ 2. If n = 2 we are done by Lemma 4.5. So assume the result for every k < n. We have Fn (G)/Fn−2 (G) = F2 (G/Fn−2 (G)) and Fn−2 (G) is lattice defined by induction if n ≥ 4. If n = 3, then Fn−2 (G) = F (G) is again lattice defined by Proposition 4.4. Then F2 (G/Fn−2 (G)) is lattice defined by Lemma 4.5, so that Fn (G) is lattice defined. In particular, it follows that if ϕ : G → G is a projectivity, then Fk (G)ϕ = Fk (G) for every k ≥ 2, as in [S2 ], Theorem 4.3.3. Corollary 4.9: Let G be a finite P -indecomposable group. Then Fn (G) is lattice defined for every n ≥ 1. Proof. This follows from Proposition 4.4 and Theorem 4.8.

5. Upper bounds Given L = (G) ∈ LF , using Theorem 3.7 we are going to derive upper bounds for the order of G, assuming that G has no cyclic direct Hall factor. Our result improves the upper bound for | G | obtained by Suzuki in [Su1 ], Theorem 1.

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M. COSTANTINI AND G. ZACHER

Isr. J. Math.

First of all we decompose L into the direct product of irreducible lattices L = L1 × · · ·× Lt , and we let G = G1 × · · ·× Gt be the corresponding direct Hall decomposition of G. If Li ∈ LP , then Gi ∈ P (ni , pi ) so that | Gi | ≤ pni i . In the remaining cases, Li ∈ L˜ and Li is not a chain by assumption. In the following ˜ In section 3 we gave a procedure to determine out we shall so assume L ∈ L. ˜ of G such that [D/1] ˜ of L a P (G)-invariant subgroup D is a di-interval, and ˜ ˜ ˜ G = D : (Z2 × Z1 × Z). Moreover, Z = D : Z2 is a P (G)-invariant Hall subgroup of G, with Z2 cyclic square-free. Suppose A is an atom of Z2 . Then A ∈ m(P1,Z˜ ) and we already observed that HA is non-empty. Let {p1 , . . . , pk } be the set of primes such that there exists H ∈ HA with H ∈ P (ni , pi ), and let p = p1 = min{p1 , . . . , pk }. Then either | A | is a divisor of each pi − 1, or | A | = p and p is a divisor of each pi − 1 for i = 2, . . . k. In particular, | A | ≤ p. Next we consider a maximal chain C, with C ≤ Z1 × Z. We recall that Z1 × Z is a cyclic Hall subgroup of G. Let α = length C, so that | C | = rα for a certain ˜ C] = 1, and also [D, ˜ C] = 1. Now prime r. Since L is irreducible, we have [Z, ˜ ˜ ˜ D is a Hall subgroup of D : C so, for each prime divisor p of | D |, there exists a ˜ which is C-invariant, i.e., Sp , C∧ D ˜ = Sp (moreover, Sylow p-subgroup Sp of D the set of C-invariant Sylow p-subgroups is a single orbit under CD˜ (C) by [A], ˜ | for which the (18.7) (2)). We consider the set πC of prime divisors p of | D ˜ action of C on the C-invariant Sylow p-subgroups Sp of D is not trivial, i.e., for which [Sp : C/1] is not reducible. So assume p ∈ πC , Sp is C-invariant. Then C acts non-trivially on Vp = Sp /Φ(Sp ), a vector space over Fp . We denote by C1 the kernel of the action of C on Vp , which coincides with C,C (Sp ): let βp be the length of C1 . Then Vp is a faithful Fp (C/C1 )-module. We introduce the set Sp of all W ≤ Vp such that W is a (faithful) simple Fp (B/C1 )-module for a certain B such that C1 < B ≤ C, and the set Tp = {dimFp W | W ∈ Sp }. Remark 5.1: The sets Sp and Tp are determined by (G). Moreover, the set Tp is the set of dimensions of all faithful simple Fp Crγ -modules, for all 1 ≤ γ ≤ α−βp . Here Crγ is the cyclic group of order rγ , α and βp are known. We shall use the following Lemma 5.1: Let p, r be fixed different primes, and let m be the smallest positive integer such that pm ≡ 1 mod r (thus m is the order of p in (Z/rZ)∗ ). Let a be such that ra pm − 1 (that is ra |pm − 1 but ra+1  pm − 1). Let X be a cyclic

Vol. 185, 2011

SUBGROUP INDEX PROBLEM

315

group of order ri and Wi a faithful simple Fp X-module (all these have the same dimension, for a fixed i). Let m(i) = dimFp Wi . Then we have 2 cases: Case 1. ra > 2 (that is r = 2, or r = 2 and p ≡ 1 mod 4). Then m(1) = · · · = m(a) = m,

m(a + k) = mrk

if k = 1, 2, . . ..

Case 2. ra = 2 (that is r = 2 and p ≡ 3 mod 4). Let b be such that 2b p2 − 1 (in particular b ≥ 3). Then m(1) = m = 1,

m(2) = · · · = m(b) = 2,

m(b + k) = 2k+1

if k = 1, 2, . . ..

Proof. See [DH], Lemma 9.6. Proposition 5.2: If Tp has at least 2 elements for a certain p ∈ πC , then r is determined. Proof. By Lemma 5.1, if we put x1 = min Tp , x2 = min Tp \ {x1 }, we get r = x2 /x1 . We now consider the situation when Tp has only one element mp for every p ∈ πC : in general, in this case we cannot determine r; anyway, we know that (G) determines p, mp and βp . Hence r is a primitive prime divisor of pmp − 1 (that is r divides pmp − 1 but not pi − 1 for 1 ≤ i < mp ). Moreover rδp , where δp = α − βp , divides pmp − 1. Out of (G) we can determine the number fp of ˜ and the number gp of Sylow p-subgroups of D ˜ which Sylow p-subgroups of D are C-invariant. We denote by Dp the set {s ∈ Π | s is a primitive divisor of pmp − 1, sδp | pmp − 1 and s|fp − gp }, and we put ΠC = p∈πC Dp . We have proved Proposition 5.3: Let C be a maximal chain of length α of Z1 × Z, | C | = rα for a certain prime r. Assume Tp = {mp } for every p ∈ πC . Then r lies in ΠC . By the previous proposition, we still can determine the order of C if ΠC has only one element. We observe that for a maximal chain C in Z we know that there are autoprojectivities σ of G for which | C σ | = | C |, so that we must have Tp = {mp } for all p ∈ πC and, if r ≥ 2 is defined as in Lemma 3.1, ΠC has at least r elements. Summarizing, we may collect in Y all the maximal chains C of Z1 for which either Tp has at least 2 elements for a certain p ∈ πC , or for which Tp has one element for every p ∈ πC but ΠC has only one element: hence [Y /1] is ˜ : Y by Corollary 2.2. Let Y1 be the complement of Y a di-interval, and such is D

316

M. COSTANTINI AND G. ZACHER

Isr. J. Math.

in Z1 , and let C1 , . . . , Ck be the maximal chains in Y1 ×Z. For every i = 1, . . . , k

k i let αi = length Ci and si = max ΠCi . Then | Y1 × Z | ≤ i=1 sα i . If A1 , . . . , Ah are the atoms of Z2 , let pi = min{p ∈ Π | there exists H ∈ HAi ∩ P (m, p)}: we have seen that | Ai | ≤ pi for each i. We have proved ˜ G not cyclic. Then Theorem 5.4: Given L = (G) ∈ L, ˜ ||Y | |G| ≤ |D

k 

i sα i

i=1

h 

pi .

i=1

From our discussion it is clear how this formula generalizes to the case when G = G1 × · · · × Gt , the Hall direct factorization of G, with each Gi irreducible, not cyclic.

References [A]

M. Aschbacher, Finite Group Theory, Cambridge University Press, Cambridge, 2000. [DFdGMS] M. De Falco, F. de Giovanni, C. Musella and R. Schmidt, Detecting the index of a subgroup in the subgroup lattice, Proceedings of the American Mathematical Society 133 (2004), 979–985. [DH] K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter, Berlin, 1992. [Ha] M. Hall, Combinatorial Theory, John Wiley & Sons, New York, 1986, [H1 ] B. Huppert, Zur Sylowstruktur aufl¨ osbarer Gruppen, Archiv der Mathematik 12 (1961), 161–169. [H2 ] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin–Heidelberg–New York, 1967. [KS] H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, Springer-Verlag, New York, 2004. [R] D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, Berlin– Heidelberg–New York, 1982. [S1 ] R. Schmidt, Verbandsisomorphismen endlicher aufl¨ osbarer Gruppen, Archiv der Mathematik 23 (1972), 449–458. [S2 ] R. Schmidt, Subgroup Lattices of Groups, W. de Gruyter, Berlin, 1994. [SZ] S. E. Stonehewer and G. Zacher, The subgroup lattice index problem, Journal of Group Theory 12 (2009), 859–872. [Su1 ] M. Suzuki, On the lattice of subgroups of finite groups, Transactions of the American Mathematical Society 70 (1951), 345–371. [Su2 ] M. Suzuki, Structure of a Group and the Structure of its Lattice of Subgroups, Springer-Verlag, Berlin–Heidelberg–New York, 1956. [Y] B. V. Yacovlev Conditions under which a lattice is isomorphic to the lattice of subgroups of a group, Algebra and Logic 13 (1974), 400–412.