Subgroups of symmetric groups: enumeration and asymptotic properties

2025/03/07 by Roney-Dougal, Colva M., Tracey, Gareth
#05A16 #20B35 #20E07 #20E25 #20F69 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2503.05416

Abstract

In this paper, we prove that the symmetric group S_n has 2^n2/16+o(n2) subgroups, settling a conjecture of Pyber from 1993. We also derive asymptotically sharp upper and lower bounds on the number of subgroups of S_n of various kinds, including the number of p-subgroups. In addition, we prove a range of theorems about random subgroups of S_n. In particular, we prove the surprising result that for infinitely many n, the probability that a random subgroup of S_n is nilpotent is bounded away from 1.

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