Maximal Subgroups of Symmetric Groups

Abstract

We show that S n has at most n 6Â11+o(1) conjugacy classes of primitive maximal subgroups. This improves an n c log 3 n bound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups of S n is of the form ( 1 2 +o(1))n. It also follows that, for large n, S n has less than n ! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at most n 17Â11+o(1) maximal subgroups of index n.

Sets of permutations that generate the symmetric group pairwise

“…, M r be a complete set of representatives of the conjugacy classes of maximal subgroups of S n of order at most ( n 5e ) n e c log n . The number of such conjugacy classes is rather small: Liebeck and Shalev [9,Corollary 4.5] proved that S n has at most ( 1 2 + o(1))n conjugacy classes of maximal subgroups, and so we find that r ( 1 2 + o(1))n. If we write [M i ] for the set of subgroups of S n that are conjugate to M i , we find that…”

Extremely primitive groups

“…A famous conjecture of G. E. Wall states that the number of maximal subgroups is at most jH j in all groups H . The result of [20] quoted in the proof of Theorem 4.7 says that Wall's conjecture holds for A n and S n for large enough n. We also know that if Soc.H / is sporadic then H satisfies Wall's conjecture. Now consider the remaining case where Soc.H / is a Lie type simple group of rank r over a field of order q.…”

“…Now for n large enough, nŠ < 2 n.n 5/=4 . Also, by [20], for n large enough A n and S n have less than nŠ maximal subgroups. Hence, for large enough n, a primitive group G D Z …”

Bounds for the probability of generating the symmetric and alternating groups

“…By [17,Page 350], there are at most 2 conjugacy classes of maximal primitive subgroups of S n of diagonal type. Hence there are at most 2 conjugacy classes of maximal primitive subgroups of A n of diagonal type.…”