Maximal subgroups of small index of finite almost simple groups

We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2<|S|$.


Introduction
All groups considered in this paper will be finite.
Given a group G, one can ask how many elements one should choose uniformly and at random to generate G with a certain given probability. The fact that an ordered r-tuple (g 1 , . . . , g r ) generates G is equivalent to the fact that {g 1 , . . . , g r } is not contained in any maximal subgroup M of G. The probability that {g 1 , . . . , g r } is contained in a maximal subgroup M of G is 1/|G : M | r . Consequently, it is of relevant interest to find good bounds for the number m n (G) of maximal subgroups of a group G of a given index n.
Note that if M is a maximal subgroup of G, then G/M G , where M G denotes the core of M in G, is a primitive group. Consequently, the proof of many results in this field relies on the subgroup structure of such groups.
According to the theorem of Baer [1] (see also [2,Theorem 1.1.7]), there are three types of primitive groups, according to whether they have a unique abelian minimal normal subgroup (type 1), a unique non-abelian minimal normal subgroup (type 2), or two non-abelian minimal normal subgroups (type 3). The theorem of O'Nan and Scott (see [2,Theorem 1.1.52]) describes the different possibilities for a primitive pair (G, U ) composed of a primitive group G of type 2 and a core-free maximal subgroup U of G. In all cases, the corresponding primitive pair is related to a primitive pair corresponding to an almost simple group with socle S, where the minimal normal subgroup of G is a direct product of copies of S. This makes crucial the study of the indices of core-free maximal subgroups of almost simple groups in the study of core-free maximal subgroups of primitive groups of type 2. Notation 1 We denote by l(X) the least degree of a faithful transitive permutation representation of a group X, that is, the smallest index of a core-free subgroup of G.
The aim of this paper is to prove that every almost simple group R with socle isomorphic to a simple group S possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index l(S) of a maximal subgroup of S or a conjugacy class of core-free maximal subgroups with a fixed index v S ≤ l(S) 2 , depending only on S. We also prove that the number of subgroups of the outer automorphism group of S is bounded by log 3 l(S) and that l(S) 2 < |S|.
These results will be applied in [3] to obtain lower bounds for the number of elements needed to generate a group with a certain probability and to obtain good lower bounds for the number of maximal subgroups of a given index of a group. They are also useful to estimate the number of possible socles of primitive groups of type 2 with a core-free maximal subgroup of a given index.
Our first main result includes relevant information over the smallest index l(S) of a maximal subgroup of a non-abelian simple group S and shows the existence of relevant subgroups of small index in an almost simple group with socle S. Moreover, we see that the order of the outer automorphism group of S is bounded by 3 log|S|. Here we reserve the symbol log to denote the logarithm to the base 2. This last bound clarifies and improves the one used in the proof of [4,Lemma 2.3], |Out S| ≤ O(log 2 n), and, as we will show in Remark 6, of order l(S) or a maximal subgroup of index v S . We present this in detail in Theorem A.
Theorem A Let S be a simple group.
1. If S does not belong to X ∪ Y, then S has a conjugacy class of ordinary maximal subgroups. In particular, if R is an almost simple group with S ≤ R ≤ Aut(S), then R has a conjugacy class of core-free maximal subgroups of index l(S). 2. If S belongs to Y, then S has at least two conjugacy classes of maximal subgroups of the smallest index l(S) and there exists a number v S ≤ l(S) 2 , depending only on S, such that if R is an almost simple group with S ≤ R ≤ Aut(S), then R has a conjugacy class of core-free maximal subgroups with index v S . 3. If S belongs to X, then S has at least two conjugacy classes of maximal subgroups of the smallest index l(S) and there exists a number v S ≤ l(S) 2 , depending only on S, such that if R is an almost simple group with S ≤ R ≤ Aut(S), then R has at least two conjugacy classes of core-free maximal subgroups with index l(S) or one conjugacy class of core-free maximal subgroups with index v S . 4. In all cases, l(S) 2 < |S| and |Out S| ≤ 3 log l(S).
Remark 1 According to [6], the automorphism group of the O'Nan simple group S ∼ = O'N has all core-free maximal subgroups of index greater than its order, so Theorem A (4) cannot be extended to the core-free maximal subgroups of almost simple groups.
Theorem B The number of subgroups of the outer automorphism group of a nonabelian simple group S is bounded by log 3 l(S). Fig. 1 Dynkin diagrams for the simple groups of Lie type Unless otherwise stated, we will follow the notation of the books [7] and [2]. Detailed information about primitive groups and chief factors of a group can be found in [2, Chapter 1].

Proofs
Our results will depend heavily on the classification of simple groups. For the simple groups of Lie type, we number the nodes of the corresponding Dynkin diagrams as in Figure 1 and denote accordingly the associated parabolic subgroups. The values of l(S) for the simple groups of Lie type have been computed in the series of papers of Mazurov [8], Vasil'ev and Mazurov [9], and Vasilyev [10][11][12]. By Lemma 2.1, if X is an almost simple group with Soc(X) ∼ = S and M is an ordinary maximal subgroup of S, then N X (M ) is a maximal subgroup of X of index |S : M |. We will use this fact without mentioning it explicitly.
Proof of Theorem A We will analyse the different possibilities for S in the classification of finite simple groups. We note that the condition l(S) 2 < |S| is equivalent to affirming that there is a maximal subgroup of S with index less than its order. We warn the reader that the information about the maximal subgroups comes from several sources and, in order to make it easier to check the results, we have preferred to adhere to the notation of the corresponding source, even if in some cases there appear some inconsistencies in the notation.

Sporadic simple groups
Suppose first that S is a sporadic simple group. It is clear that if the outer automorphism group of S is trivial, then S / ∈ X ∪ Y and the result is trivially valid. In the other cases, the outer automorphism group has order 2. In the sporadic simple groups M 22 , J 2 , Suz, HS, McL, He, Fi 22 , HN, and J 3 , according to the Atlas [6], the largest maximal subgroups are ordinary and so l * (A) = l(S). The maximal subgroups of the group Fi ′ 24 and its automorphism group Fi 24 have been studied in [13]. The smallest index maximal subgroup Fi 23 is ordinary. In the Mathieu group M 12 , there are two classes of the smallest index maximal subgroup, with index 12 and there is a class of ordinary maximal subgroups of index 144 = 12 2 (see [6]). In the O'Nan group S ∼ = O ′ N, according again to [6], there are two conjugacy classes of maximal subgroups of type L 3 (7) : 2, of the smallest index 122 760, fused under the outer automorphism, giving a conjugacy class of novel maximal subgroups of type For all the sporadic groups S, we also see that the inequality l(S) 2 < |S| holds for all groups whose maximal subgroups have been described in [6]. The conclusion for the Baby Monster group B follows, by [14], for the smallest index maximal subgroup 2. 2 E 6 (2) : 2. The conclusion for the Monster group M is also true, by [15,16], with the maximal subgroup 2.B. Finally, it is clear that |Out S| ≤ 2 ≤ log l(S) for all sporadic simple groups S.

Unitary groups
Suppose now that S ∼ = PSUn(q) with n ≥ 3 and q > 2 if n = 3; then |Out S| ≤ n · f · 1 with p f = q 2 . The smallest index of a maximal subgroup of S is given in [8,Theorem 3].
We consider first the case S ∼ = PSU 3 (5). We see in [6] that the automorphism group of S is isomorphic to S 3 and that there are three conjugacy classes of maximal subgroups of the smallest possible index, of type Alt(7) and index l(S) = 50 and order 2 520. Moreover, there is an ordinary maximal subgroup of type 5 1+2 + : 8 and index 126 ≤ 50 2 . In this case, |Out S| = 3 · 2 · 1 = 6 ≤ 3 log l(S).

Orthogonal groups
Suppose now that S ∼ = O ε n (q) is an orthogonal group with n ≥ 7, n even if q = 2 f . The smallest index maximal subgroups of S have been described in [9,Theorem].
Groups of type G 2 (q) The maximal subgroups of smallest index of the simple groups G 2 (q), q > 2, have been studied in [10,Theorem 1].
Assume that S ∼ = G 2 (3). Then P ∼ = PSU 3 (3) : 2 is a maximal subgroup of the smallest possible index l(S) = 351 and order 12 096 > l(S). By [6], there are two conjugacy classes of maximal subgroups of this index and there is a conjugacy class of ordinary maximal subgroups of type PSL 2 (8) : 3 and index 2 808 < l(S) 2 . In this case, |Out S| = 2 < log l(S).
In [12,Theorem 2], it is shown that if S ∼ = 2 G 2 (q), with q = 3 f and f an odd integer greater than 1, there is a class of smallest index maximal subgroups isomorphic to P ∼ = (3 f .3 f .3 f ) : (q − 1) and index q 3 + 1. By [17, Table 8.43], these subgroups are ordinary and, clearly, l(S) < |P | and |Out S| = f ≤ log q ≤ log l(S).

Remark 2
We thank one of the anonymous referees for drawing our attention to the interesting paper [21] of Alavi and Burness. These authors have obtained in their Theorems 2-5 for each simple group G and in their Theorem 7 for each almost simple group G the list of all maximal subgroups H of G with |H| 3 ≥ |G|. They call them large. In fact, all maximal subgroups appearing in the proof of Theorem A are large in this sense and so all of them are mentioned in [21].
Remark 3 Note that the smallest index of a smallest core-free maximal subgroup of an almost simple group with socle PSLn(q) with n ≥ 3 can be different from the indices of the parabolic and the double parabolic subgroups. According to [6], if S = PSL 3 (4), then the extension S.2 1 contains a maximal subgroup of type M 10 and least index 56, different from the indices of the parabolic subgroups of type P 1 , of index 21, that do not exist in this extension, and the double parabolic subgroups of type P 1,2 , of index 105, that also appear as a maximal subgroup of S.2 1 .

Remark 5
The groups PSLn(q) for n ≥ 3, q = q 2 0 , q 0 a prime power, contain always a maximal subgroup of the form PSLn(q 0 ) or PSUn(q 0 ), but their indices in PSLn(q) are polynomials on q 0 of degree larger than the degree of l(PSLn(q)) 2 = (q 2n 0 − 1) 2 /(q 2 0 − 1) 2 when n ≥ 4. Hence this construction cannot be extended further. The groups PSL 2m (q) for m ≥ 2, q a prime power, (2m, q) = (4, 2), contain a parabolic subgroup Pm that is ordinary. However, for m ≥ 4, this subgroup has index in PSL 2m (q) that is a polynomial on q of degree larger than the one of l(PSL 2m (q)) 2 = (q 2m −1) 2 /(q−1) 2 . This justifies that these constructions cannot be extended further and so PSL 3 (q 2 0 ), PSL 4 (q), PSL 6 (q) belong to the class Y, but not the linear groups in larger dimensions.
it follows that the bound |Out S| ≤ 3 log l(S) cannot be improved.

Proof of Theorem B
The result is clear for sporadic and alternating groups, since then the outer automorphism group is trivial, isomorphic to C 2 , or isomorphic to C 2 × C 2 . It only remains to consider the case when S is a simple group of Lie type. According to [6, Table 5], the outer automorphism group is isomorphic to an extension of a metacyclic group by a cyclic group, with the possible exception of O + 2m (q) with m ≥ 4, m even. By [22, page 181], If m ≥ 6 is even, the same arguments show that By considering the normal subgroup isomorphic to C f , we see that all subgroups of Out(O + 2m (q)) for m ≥ 4, m even, can also be generated by at most 3 generators. In all other cases, all subgroups of Out S are extensions of a metacyclic group by a cyclic group and so they are also 3-generated.
If |Out S| ≤ log l(S), then given a subgroup of G, we have at most log 3 l(S) possibilities for a generating set. It follows that the number of subgroups of Out S is at most log 3 l(S). Therefore we must study the cases in which the inequality |Out S| ≤ log l(S) can fail, that is, the cases mentioned in Theorem A (5).
Note that Out S and all its subgroups are 2-generated. One of the generators can be taken in the subgroup of order 2, while the other one can be chosen in 2f different ways. This gives for the number of subgroups an upper bound of 4f ≤ 2 · 2 log 3 log l(S) = 4 log 3 log l(S).
Since 4/log 3 ≤ log 2 5 ≤ log 2 l(S), we conclude that the number of subgroups of Out S is bounded by log 3 l(S).
Suppose that f = 1. Since l(PSL 3 (2)) = 7 and |Out PSL 3 (2)| = 2, and l(PSL 4 (2)) = 8 and |Out PSL 4 (2)| = 2, we can assume that (m, q) / ∈ {(3, 2), (4, 2)}. Consequently, l(S) = (q m − 1)/(q − 1) = q m−1 + q m−2 + · · · + q + 1 and so log l(S) > (m − 1) log q. We have that Out S is 2-generated and has order gcd(m, q − 1) · 2. Moreover, all subgroups of Out S are 2-generated, the first generator can be taken in the normal cyclic subgroup of order gcd(m, q − 1) and the second one in Out S. This gives at most gcd(m, q − 1) 2 · 2 possibilities for a subgroup of Out S. Since , (4, 2)}. Hence we can assume that f ≥ 2. Every subgroup of Out S is 3-generated, and the generators can be taken one in the group of diagonal automorphisms of order gcd(m, q − 1), another one in the group generated by the diagonal and field automorphisms of order gcd(m, q −1)f , and the third one in Out S. It follows that the number of possible choices is at most 2f 2 gcd(m, q−1) 3 . Suppose first that gcd(m, q−1) < m, then gcd(m, q − 1) ≤ m/2 and so the number of possible subgroups is bounded by Therefore we can assume that gcd(m, q − 1) = m, that is, m | q − 1. The number of possible subgroups of Out S is bounded by If p ≥ 3, then m 3 < (m−1) 3 log 3 p and so the number of possible subgroups of Out S is again bounded by log 3 l(S). Therefore we can suppose that p = 2. In particular, m must be odd. Assume that m is a prime. Then the number of choices of the element of the group of diagonal automorphisms can be reduced from m to 2, namely the trivial element and a generator. This gives that the number of possible subgroups of Out S is bounded by If m = 5, then l(S) ≥ (2 4 ) 4 = 2 16 , and so log l(S) ≥ 16. It follows that 4m 2 /(m − 1) 2 < log l(S). If m = 7, then l(S) ≥ (2 3 ) 6 = 2 18 and so log l(S) ≥ 18. Consequently, 4m 2 /(m − 1) 2 < log l(S). Hence for m ∈ {5, 7}, the number of subgroups of Out S is bounded by log 3 l(S). Assume now that m ≥ 9 is odd. The number of choices of the element of the group of diagonal automorphisms can be reduced to the number of subgroups of this cyclic group, which coincides with the number of divisors of m. Since m is odd, this number is not greater than 2m/3. The number of possible choices for the generators of a subgroup of Out S is bounded by (2m/3) · mf · 2mf = 1 6f (2mf ) 3 ≤ 1 12 8m 3 (m − 1) 3 log 3 l(S), and 8m 3 / 12(m − 1) 3 ≤ 243/256, so that this number is bounded by log 3 l(S). It only remains the case m = 3. In this case, the group of outer automorphisms of S has the presentation Out S = x, y, z | x 3 = y f = z 2 = 1, x y = x −1 , x z = x −1 , y z = y −1 .
Note that y 2 centralises x . Now where a =x, b =z, c =ȳz. Note that every subgroup of Sym(3)×C 2 is 2-generated. A pair of generators can be obtained by taking an element of a, b and an element of the set {1, c, ac, bc, abc, a 2 bc}. The preimages of these sets under the natural epimorphism from Out S onto (Out S)/ y 2 have 6(f /2) = 3f elements each. Hence every element of Out S can be obtained by considering an element of y 2 , for which we have f /2 choices, and the 3f choices for each element of the preimages. This gives a bound for the number of subgroups of (f /2)(3f ) 2 = 9f 3 /2 = 9(2f ) 3 /16 < (9/16) log 3 l(S) < log 3 l(S). This completes the proof for the linear case.
reading carefully the manuscript and for identifying several typos and incorrect statements. Their contributions have improved considerably the presentation of the results of this paper and their proofs. We also thank the Editor-in-Chief, Professor Fernando Etayo, for his kind help in the revision of the manuscript and for his empathy and his patience with our concerns.

Declarations Funding
These results are part of the R+D+i project supported by the Grant PGC2018-095140-B-I00, funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe", as well as by the Grant PROMETEO/2017/057 funded by GVA/10.13039/501100003359, and partially supported by the Grant E22 20R, funded by Departamento de Ciencia, Universidades y Sociedad del Conocimiento, Gobierno de Aragón/10.13039/501100010067.