The diameter of products of finite simple groups

Daniele Dona

doi:10.26493/1855-3974.2614.44c

Abstract

Following partially a suggestion by Pyber, we prove that the diameter of a product of non-abelian finite simple groups is bounded linearly by the maximum diameter of its factors. For completeness, we include the case of abelian factors and give explicit constants in all bounds.

Highlights

An important area of research in finite group theory in the last decades has been the production of upper bounds for the diameter of Cayley graphs of such groups
For any finite group G, the maximum diameter over all Cayley graphs defined by symmetric sets of generators of G is called the diameter of G
The results are essentially in line with what is generally expected from the behaviour of the diameter of finite groups

Summary

Introduction

An important area of research in finite group theory in the last decades has been the production of upper bounds for the diameter of Cayley graphs of such groups. Dependence on the maximum of the diameter of the components, instead of dependence on their product as Schreier’s lemma (see Lemma 2.1) would naturally give us, was already established in [2, Lemma 5.4]: in that case, the diameter was bounded as O(d2), where the dependence of the constant on n was polynomial as in our statement This result was improved in [5, Lemma 4.13] to O(d), but only in the case of alternating groups: this was done in part to fix a mistake in the use of the previously available result in Babai-Seress, which is why only alternating groups were considered, as permutation subgroups were the sole concern in both papers; a suggestion by Pyber, reported in Helfgott’s paper, points at the results by Liebeck and Shalev [8] as a way to prove a bound of O(d) for a product of arbitrary non-abelian finite simple groups

Dona: The diameter of products of finite simple groups

Preliminaries

Proof of the main theorem

Concluding remarks