Abelian sections of the symmetric groups with respect to their index

Luca Sabatini

doi:10.1007/s00013-021-01667-0

Abstract

We show the existence of an absolute constant alpha >0 such that, for every k ge 3, G:= mathop {mathrm {Sym}}(k), and for every H leqslant G of index at least 3, one has |H/H'| le |G:H|^{alpha / log log |G:H|}. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.

Highlights

Abelian quotients of permutation groups attracted attention for the first time in [4], where the authors show that an abelian section of Sym(k) has order at most 3k/3 for every k ≥ 3
Better bounds hold for primitive groups [1], and for transitive groups [5]
A different aspect concerning the subgroups of the symmetric groups is revealed: as the index increases, the abelian quotients grow as slowly as possible

Summary

Introduction

Abelian quotients of permutation groups attracted attention for the first time in [4], where the authors show that an abelian section of Sym(k) has order at most 3k/3 for every k ≥ 3. Better bounds hold for primitive groups [1], and for transitive groups [5]. In these notes, a different aspect concerning the subgroups of the symmetric groups is revealed: as the index increases, the abelian quotients grow as slowly as possible. |H/H | ≤ |G : H| α/ log log |G:H| This bound is sharp in a number of situations, which make the proof by induction somewhat challenging.

Abelian sections of the symmetric groups

It remains to prove that this is at least