Abelian sections of the symmetric groups with respect to their index

Abstract

We show the existence of an absolute constant $\alpha >0$ such that, for every $k\ge 3$, $G:=\mathrm{Sym}(k)$, and for every $H⩽G$ of index at least $3$, one has $|H/[H,H]|\le |G:H{|}^{\alpha /\log \log |G:H|}$. 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.

arXiv comment6 pages

Cited by in corpus (1)

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