Bounds on the number of maximal subgroups of finite groups

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability.


Introduction
All groups considered in this paper will be finite.
The fact that an r-tuple of elements of a group G does not generate G is equivalent to the fact that all these elements belong to a certain maximal subgroup M of G. The probability that a certain r-tuple of elements of G chosen at random and with a uniform probability distribution is contained in a maximal subgroup M of G is 1/|G : M| r . This makes interesting to consider the number m n (G) of maximal subgroups of a group G of index n for a natural number n in problems related with the probability that a certain tuple of elements of the group generates it.
Pak [Pak], motivated by potential applications for the product replacement algorithm, widely used to generate random elements in a finitely generated group, analysed an invariant related to the probability a tuple generates a group.
Definition 1.1. Given a group G, we denote by V(G) the least positive integer k such that G is generated by k random elements with probability at least 1/e. Here and in all this paper, the symbol log will be used to denote the logarithm to the base 2, and we follow the convention that log 0 = −∞.
Given a maximal subgroup M of a group G, the quotient G/M G of G by the normal core M G of M in G is a primitive group. According to a wellknown theorem of Baer [Bae57] (see also [BBE06, Theorem 1.1.7]), G/M G is of one of three different types. In this case, we say that the maximal M is of the same type as the primitive group G/M G it induces, like in [BBE06, Definition 1.1.19]. Lubotzky [Lub02] established bounds on the number of maximal subgroups of type 1 of a given index n and on the number of maximal subgroups of types 2 and 3 of index n by considering the number of abelian and non-abelian chief factors, respectively, in a given chief series of the group. He obtained the following result that confirms the validity of Pak's conjecture.
Theorem 1.2 (Lubotzky, [Lub02, Corollary 2.6]). If G is a group with r chief factors in a given chief series, r a of them abelian and r b of them non-abelian, then m n (G) ≤ 1 2 (r b + 1)r b + r a n d(G) n 2 ≤ r 2 n d(G)+2 . Furthermore, V(G) ≤ 1 + log log|G| log i(G) + max d(G), log log|G| log i(G) + 2.02, where i(G) denotes the smallest index of a proper subgroup of G.
The bound for V(G) in Theorem 1.2 depends on the following invariant. Here we use log n x to denote the logarithm to the base n of x, that is, log n x = log x/ log n = ln x/ ln n. There is a close relation between M(G) and V(G).
The proof of the left hand side bound in the above result depends on the following result.
Theorem 1.5 (see [Lub02,Theorem 1.3] or [DL06,Theorem 21]). There exists a constant b such that for every group G and every n ≥ 2, G has at most n b core-free maximal subgroups of index n. In fact, b = 2 will do.
Unless otherwise stated, we will follow the notation of the books [DH92] and [BBE06]. Detailed information about primitive groups and chief factors, crowns, and precrowns of a group can be found in [BBE06, Chapter 1].
Detomi and Lucchini [DL03], with the help of the number λ(G) of non-Frattini chief factors in a given chief series of G and by considering their associate crowns, proved independently also the validity of Pak's conjecture.
Theorem 1.6 (Detomi and Lucchini [DL03,Theorem 20]). There exists a constant c such that, for any group G, where λ(G) denotes the number of non-Frattini chief factors in a given chief series of G.
Here the symbol ⌊x⌋ denotes the defect integer part of x, that is, the largest integer number n such that n ≤ x.
The proof of this result uses some results of Dalla Volta and Lucchini [DVL98] and Dalla Volta, Lucchini, and Morini [DVLM01] about the number of generators of powers of certain diagonal-type subgroups of direct powers of copies of a monolithic group.
The aim of this paper is to obtain tighter bounds for m n (G), and so for V(G), by considering the numbers of maximal subgroups of each type, as in Lubotzky's paper [Lub02], and with the help of the crowns associated to abelian chief factors, like in Detomi and Lucchini's paper [DL03], and to non-abelian chief factors. Our bounds will depend on four invariants. Our first invariant is related to maximal subgroups of type 1.
Definition 1.7. Let G be a group and let n > 1 be a natural number. We denote by cr A n (G) the number of crowns associated to complemented abelian chief factors of order n of G, that is, the number of G-isomorphism classes of complemented abelian chief factors of G.
Clearly, cr A n (G) = 0 unless n is a power of a prime. In order to define our second invariant, which concerns non-abelian chief factors and is related to the primitive quotients of type 2, we must recall the next definition. Definition 1.9. Let n be a natural number. The symbol rs n (G) denotes the number of non-abelian chief factors A in a given chief series of G such that the associated primitive group [A] * G has a core-free maximal subgroup of index n.
Our third and fourth invariants concern also non-abelian chief factors and contain information about the primitive quotients of type 3 of the group.
Definition 1.10. Let n be a natural number. The symbol ro n (G) denotes the number of non-abelian chief factors A in a given chief series of G such that A has order n.
For the definition of our fourth invariant, the following auxiliary invariant is useful.
Definition 1.11. Let G be a group. For a characteristically simple group A, that is, a direct product of copies of a simple group S, r A (G) denotes the number of non-Frattini chief factors isomorphic to A in a given chief series of G.
According to Theorem A of [BBERJS], r A (G) coincides with the largest number r such that G has a normal section that is the direct product of r non-Frattini chief factors of G that are isomorphic (not necessarily G-isomorphic) to A. Definition 1.12. Let n be a natural number. The symbol rm n (G) denotes the maximum of the lenghts of the G-crowns associated to non-abelian chief factors of order n of G.
We will see that, according to Theorem 2.6, the non-abelian chief factors A of G of order n fall into at most two isomorphism classes. We also note that a non-abelian chief factor can be counted in rs n (G) for different values of n, if the corresponding primitive group has core-free maximal subgroups of different indices. Furthermore, a non-abelian chief factor of order n that is counted in ro n (G) can also be counted in rs n (G) if the primitive group associated to the corresponding chief factor has a maximal subgroup complementing its socle (see [JSL98] or [BBE06, Theorem 1.1.48]), We obtain a bound for m n (G) that improves the one of Theorem 1.2.
Theorem A. Let G be a non-cyclic group with r chief factors in a given chief series. For every natural n ≥ 2, m n (G) ≤ rn d(G)+2 .
This bound will appear as a consequence of a tighter bound in terms of the invariants cr A n (G), rs n (G), rm n (G), and ro n (G) presented in Theorem 2.10. We also obtain a bound for V(G).
where η(G) is a function bounded by a linear combination of d and the maxima of log n cr A n (G), log n rs n (G), log n ro n (G), and log n rm n (G). The precise value of the function η(G) will be presented in detail in Theorem 2.11.
We obtain upper bounds for the number of maximal subgroups of a given index in a group in Section 2. We show that the value of the constant in the bounds of Detomi and Lucchini [DL03] (Theorem 1.6) can be computed from Theorem B. This will be done in Section 3. We also present a general procedure (Construction 3.3) to obtain, starting from a d-generated primitive group H of type 1, a d-generated group G with all possible G-isomorphism classes of chief factors isomorphic to the socle of H and with associated primitive groups isomorphic to H. Similar ideas (see Remark 3.5) can be used to create a d-generated group with all possible crowns associated to non-Frattini abelian chief factors of a given order whose associated primitive groups are d-generated.
2 Upper bounds for the number of maximal subgroups of a given index in a group 2.1 Bounds for the number of maximal subgroups of type 1 Let L be a non-cyclic group with a unique complemented abelian minimal normal subgroup N. Assume also that N is complemented in L. As in [DVL98], let L k be the direct product of k copies of L and set Then L k is the product of N k and a diagonal subgroup Let us call f (d) = k + 1. Then, by [DVL98], Theorem 2.1. Let U be a maximal subgroup of type 1 of a d-generated group G and let n = |G : U|. If U is not normal in G, then the number of maximal In all cases, this number is less than or equal to n d − 1.
Proof. Consider the crown C/R associated to A, Observe that C/U G = Soc(G/U G ). If . By the comments after [DVL98, Theorem 2.7], we have that Moreover, there exists an isomorphism ϕ : G/M G −→ L satisfying that the composition with the projection over G/C, G/M G −→ L −→ G/C, has kernel C and gives he identity on G/C. Let us now give a bound for the number of possible kernels of epimorphisms from L K to L giving the identity on G/C. By [DVL98, Lemma 2.5], the number of possible kernels of epimorphisms from L k onto L giving the identity on G/C is (q k − 1)/(q − 1). By Equation (1), The number of maximal subgroups of type 1 with a given core coincides with the number of complements of the minimal normal subgroup of the associated primitive group, namely n|H 1 (L/A, A)|. Hence the number of Suppose now that A is central, that is, that U is normal in G = C, then n is a prime and G/R = C/R is a direct product of k cyclic groups isomorphic to C n . Since G is d-generated, k ≤ d. In this case, q = n is a prime number and the number of normal subgroups M of index n such that R ≤ M ≤ G is Remark 2.2. The bound of Theorem 2.1 is attained in groups like L k , where L ∼ = Sym(4). In this group, q = 2, n = 4, and |H 1 (G/C, A)| = 1. If k = f (d) − 1 = 2d − 2, for d ≥ 2, we obtain that the number of maximal subgroups of index 4 of L k is exactly Corollary 2.3. The number of maximal subgroups M of type 1 and index n = p r of a d-generated group G is less than or equal to (n d − 1)cr A n (G).
Proof. Given a maximal subgroup U of G of type 1 and index n, we have that its socle Soc(G/U G ) must fall into one of the G-isomorphism classes of chief factors. By Theorem 2.1, we obtain that the number of maximal subgroups M of G such that Soc(G/M G ) ∼ = G Soc(G/U G ) is bounded by n d − 1. Since the number of G-isomorphism classes of chief factors of order n is cr A n (G), we obtain that the number of maximal subgroups of type 1 and index n of G must be bounded by (n d − 1)cr A n (G).

Bounds for the number of maximal subgroups of type 2
The next result provides a bound for the number of maximal subgroups of type 2 of a given index of a group. and another chief factor G-isomorphic to Soc(G/U G ). Consequently the number of primitive quotients of G associated to a maximal subgroup of index n is rs n (G). By Theorem 1.5, the number of core-free maximal subgroups of index n of G/U G for a maximal subgroup U of type 2 of G is bounded by n 2 , that is, the number of maximal subgroups of index n of G with core U G is bounded by n 2 . Hence the number of maximal subgroups of G of type 2 and index n is bounded by rs n (G)n 2 .

Bounds for the number of maximal subgroups of type 3
Now we analyse the maximal subgroups of type 3. The following two results show that the minimal normal subgroups of primitive groups of type 3 fall in at most two isomorphism classes. Although this is not needed for the proofs of our results, we include this information here for the sake of completeness.
The first result appears as a consequence of the classification of simple groups.
Lemma 2.5. The number of non-abelian simple groups of a given order is at most 2.
A generalisation of this result to characteristically simple groups was formulated as a question by Cameron [Cam81], who attributed its proof to Teague in Note (ii) at the end of his paper.
Theorem 2.6 (see [KLST90, Theorem 6.1]). Let S and T be non-isomorphic finite simple groups. If |S a | = |T b | for some natural numbers a and b, then a = b and S and T are either PSL 3 (4) or PSL 4 (2), or are O 2n+1 (q) and PSp 2n (q) for some n ≥ 3 and some odd q.
As in [JSL98], we use the non-abelian cohomology with the terminology of Serre in [Ser64]. Given a G-group B and a 1-cocycle β ∈ Z 1 (G, B), then b η(g) = b gg β defines a homomorphism η : G −→ Aut B. The corresponding G-group is denoted by B β and called the G-group obtained from B by torsion via β.
Lemma 2.7. Let G be a d-generated group. Let n be a natural number which is a power of the order of a non-abelian simple group. Then rm n (G) ≤ n d .
Proof. Consider a crown with a chief factor A of order n and let B be another chief factor in the same crown as A. We have that A and B are G-connected (see [BBE06, Definition 1.2.5]), but not G-isomorphic. According to [JSL98, Proposition 1.2], there exists a 1-cocycle β ∈ Z 1 (G, B) such that A is isomorphic to B β as G-groups, that is, there exists an isomorphism ϕ : A −→ B such that a gϕ = a ϕgg β for each a ∈ A and g ∈ G. Hence the number of chief factors G-connected to B but not G-isomorphic to B is bounded by |Z 1 (G, B)| and so rm n (G) ≤ |Z 1 (G, B)|. Since these 1-cocycles are uniquely determined by the image of a generating system of G, we have that the number of cocycles is at most n d . It follows that rm n (G) ≤ n d .
Theorem 2.8. Let G be a d-generated group and let n be a natural number that is a power of the order of a non-abelian simple group. Assume that G has s n crowns of (non-abelian) chief factors of order n that are the products of t 1 , t 2 , . . . , t sn chief factors, respectively. The number of maximal subgroups of G of type 3 and index n is bounded by Proof. Given a maximal subgroup U of G of type 3 and index n, the quotient G/U G is a primitive group of type 3. The socle of a primitive quotient of type 3 of G is a direct product of two minimal normal subgroups that are G-isomorphic to two G-connected chief factors of G belonging to the same G-crown. Since the core of this maximal subgroup is the intersection of the centralisers of the minimal normal subgroups of the quotient, that are the centralisers of the chief factors appearing in the crown, we have that the number of choices for the core is t i (t i − 1)/2, where t i is the number of chief factors in the corresponding crown. This shows that the number of cores of such maximal subgroups is bounded by sn j=1 t j (t j − 1)/2. By Theorem 1.5, each of the quotients by these cores has at most n 2 core-free maximal subgroups of index n. The result follows.
Corollary 2.9. Let G be a d-generated group and let n be a natural number which is a power of the order of a non-abelian simple group. The number of maximal subgroups of G of type 3 and index n is bounded by where s n is the number of G-crowns of G associated to non-abelian chief factors.
Proof. Since, with the notation of Theorem 2.8, we have that ro n (G) = sn i=1 t i and rm n (G) = max sn i=1 t i , this result is an immediate consequence of Theorem 2.8.

General bounds
Denote by T the set of all prime powers greater than 1 and by S the set of all powers of the orders of non-abelian simple groups. As a consequence of Corollary 2.3, Theorem 2.4, and Theorem 2.8, we obtain the following conclusion.
Theorem 2.10. The number m n (G) of maximal subgroups of index n of a d-generated group G satisfies the following bounds: 1. If n ∈ T, then m n (G) ≤ (n d − 1)cr A n (G) + n 2 rs n (G).
We can use the bound A + B ≤ 2 max{A, B} to obtain the following bounds for m n (G): 1. If n ∈ T, then m n (G) ≤ 2 max{n d cr A n (G), n 2 rs n (G)}.
Therefore we can obtain the following bound for M(G).
≤ max n max{d + log n 2 + log n cr A n (G), 2 + log n 2 + log n rs n (G) | n ∈ T}, 2 + max log n 2 + log n rs n (G), log n rm n (G) + log n ro n (G) | n ∈ S , 2 + max{log n rs n (G) | n / ∈ S ∪ T} ≤ max d + max n∈T {log n 2 + log n cr A n (G)}, 2 + max n {log n 2 + log n rs n (G)}, 2 + max n∈S log n rm n (G) + log n ro n (G)} ≤ max d + max n∈T {log n 2 + log n cr A n (G)}, 2 + max n {log n 2 + log n rs n (G)}, We are in a position to prove Theorem A.
Proof of Theorem A. Note that cr A n (G) ≤ r, rs n (G) ≤ r, ro n (G) ≤ r, and rm n (G) ≤ n d . By Theorem 2.10, if n ∈ T, then m n (G) ≤ n d r + n 2 r ≤ 2n d r ≤ n d+2 r; if n ∈ S, then m n (G) ≤ n 2 r + n 2 rn d /2 ≤ n d+2 r, and if n / ∈ S ∪ T, then m n (G) ≤ n 2 r ≤ n d+2 r.
We can obtain from Theorem 2.10 the value of the function of the upper bound of Theorem B. Remark 2.12. The terms log n 2 in Theorem 2.11 depend on the bound A + B ≤ 2 max{A, B}. If A = 0 or B = 0, then A + B = max{A, B}. In particular, in the computation of the maxima the terms log n 2 appear only if n ∈ T and cr A n (G)rs n (G) = 0, or n ∈ S and rs n (G)ro n (G) = 0. In all other cases, log n 2 can be safely removed to estimate V(G).

Discussion
The aim of this section is to show that the value of the constant in the upper bound of Theorem 1.6 appears as a consequence of Theorem B and to present a construction of groups with a large number of crowns of chief factors whose associated primitive quotients are isomorphic to a given primitive group of type 1.

Determination of the constant of Theorem 1.6
We can use the upper bound of Theorem B to determine the value of the constant c of Theorem 1.6.
Theorem 3.1. The bounds of Theorem 1.6 hold with c = 5.02.
Proof. Suppose first that the number λ(G) of non-Frattini chief factors in a given chief series of G is 1. In this case, G/Φ(G) is the only non-Frattini chief factor of G and G/Φ(G) must be simple, either abelian or non-abelian. If G/Φ(G) is abelian, then G is cyclic and G has exactly one crown of non-Frattini abelian chief factors. Therefore V(G) ≤ d + log 2 2 + 2.02 = d + 3.02 (note that the term log 2 2 can even be safely removed from this sum). Assume now that G/Φ(G) is non-abelian. It follows that rs n (G) = 1, where n is the index of a maximal subgroup of G/Φ(G) and so V(G) ≤ 4.02 + log 5 2 < 5.02 (we could even remove the term log 5 2). In both cases, V(G) ≤ ⌊d + 3.02⌋. Now assume that λ(G) > 1. It will be enough to show that for a constant c, all three elements whose maximum is being considered are bounded by d + c log λ(G) for a constant c. Note also that cr A n (G), rs n (G), ro n (G), and rs n (G) are less than or equal to λ(G). The first term satisfies d + 2.02 + max n {log n 2 + log n cr A n (G)} ≤ d + 2.02 + 1 + log λ(G) ≤ d + 4.02 log λ(G).

Some examples and constructions
We conclude this section by presenting a couple of examples in which our bound improves substantially the previously known bounds. In the following we will construct some groups with many crowns of chief factors whose associated primitive quotients are isomorphic to a given monolitic primitive group. This construction will depend on a a general construction for subdirect products.
We have that the restriction toĜ of the natural projection π i : D −→ G i , 1 ≤ i ≤ r, is a group epimorphism with kernelĜ ∩ k =i G k . Hence each chief factor H i /K i of G i is isomorphic to a chief factor H/K ofĜ. Furthermore, G i acts on H i /K i asĜ acts on H/K. Moreover, r i=1 (Ĝ ∩ k =i G k ) = 1, and soĜ is a subdirect product of the subgroups G i . Consequently, each chief factor ofĜ is isomorphic to a chief factor of one of the G i . However, it might happen that the number of chief factors ofĜ is less than the number of chief factors of all the G i . For instance, this happens if G i = a 1i ∼ = C p , 1 ≤ i ≤ 2: in this case,Ĝ ∼ = C p .
The following construction for d-generated groups with all possible crowns whose associated primitive quotients are isomorphic to a given monolitic primitive group is motivated by an argument of Hall [Hal36].
Construction 3.3. Let G be a d-generated group. We can consider the set Ω of all generating ordered d-tuples of G. The group Aut G acts on Ω in the natural way. This decomposes Ω into orbits for the action of Aut G, Ω = Ω 1 ∪ · · · ∪ Ω r say. Let (a i1 , . . . , a id ) be an element of the orbit Ω i , 1 ≤ i ≤ r. We can apply Construction 3.2 to G i = a i1 , . . . , a id = G, 1 ≤ i ≤ r, to obtainĜ = a 1 , . . . , a d . If G is a primitive group of type 1 and socle N, then the socle ofĜ is the product of all faithful and irreducible modules for G/N such that the corresponding primitive group is isomorphic to G.
An application of this construction with GAP [GAP21] shows that the other values of the ranks and the numbers of abelian crowns are zero. We note that if 1 ≤ i < j ≤ 3, two chief factors of S corresponding to chief factors ofĜ i andĜ j , respectively, cannot be S-isomorphic, because the groups of automorphisms induced by S on the chief factors in the primitive groups G i and G j are not isomorphic. Therefore cr A 3 (S) = 1, cr A 7 (S) = 9, cr A 8 (S) = 146, r GL 3 (2) (S) = 57; the other values are zero (we note that, in fact, S coincides with the direct productĜ 1 ×Ĝ 2 ×Ĝ 3 ). Since the indices of the maximal subgroups of GL 3 (2) are 7 and 8 (see for instance [CCN + 85]), we conclude that rs 7 (S) = rs 8 (S) = 57 and ro 168 (S) = rm 168 (S) = 57. The crowns of chief factors of order 8 inĜ 1 are minimal normal subgroups, inĜ 2 are products of three minimal normal subgroups, while inĜ 3 are products of two minimal normal subgroups. The crowns of chief factors of order 7 in G 1 andĜ 2 coincide with the corresponding chief factors. There is a unique crown composed of two central chief factors of order 3. Hence S has 16 + 16 · 3 + 114 · 2 = 292 chief factors of order 8, 8 chief factors of order 7, 2 chief factors of order 3, and 57 chief factors isomorphic to GL 3 (2).