Submission date: 25 June 2021

Abstract:

Given a finite group G and a generating set S ⊆ G, the diameter diam(G,S) is the least integer n such that every element of G is the product of at most n elements of S. In this paper, for bounded |S|, we characterize groups with polynomial diameter as the groups with a large abelian section close to the top, precisely of size an exponential portion of the size of the full group. This complements a key result of Breuillard and Tointon. As a consequence, groups with polynomial diameter have many conjugacy classes, and contain a large nilpotent subgroup of class at most 2.

Mathematics Subject Classification: 20F69

Keywords and phrases:

Full text arXiv 2106.13577: pdf, ps.