Diameter bounds for arbitrary finite groups and applications

TL;DR

This paper establishes a general bound on the diameter of finite groups based on their composition factors and abelian sections, with various applications to soluble and anabelian groups, and implications for longstanding conjectures.

Contribution

It introduces a universal diameter bound for finite groups depending on composition factors and abelian sections, advancing understanding of group diameters and related conjectures.

Findings

Finite soluble groups of exponent e have diameter e ( ( |G|)^8

Anabelian groups with bounded-rank composition factors have polylogarithmic diameter

Transitive soluble subgroups of S_n have diameter n^5

Abstract

We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if $G$ is a finite soluble group of exponent $e$, $\mathrm{diam}(G) \ll e (\log |G|)^8$, (2) anabelian groups with bounded-rank composition factors have polylogarithmic diameter, (3) transitive soluble subgroups of $S_n$ have diameter $\ll n^5$, and (4) Grigorchuk's gap conjecture holds for any finitely generated group acting faithfully on a bounded-degree rooted tree. Additionally, conditional on Babai's conjecture, (5) any transitive permutation group of degree $n$ has diameter bounded by a polynomial in $n$ (a folkloric conjecture), and (6) Grigorchuk's gap conjecture holds for residually finite groups, and thus the conjecture reduces to the simple case.