On finite groups with polynomial diameter

TL;DR

This paper characterizes finite groups with polynomial diameter in terms of their large abelian sections and nilpotent subgroups, providing insights into their structure and conjugacy class distribution.

Contribution

It offers a new structural characterization of groups with polynomial diameter, linking them to large abelian sections and nilpotent subgroups, complementing prior results.

Findings

Groups with polynomial diameter have large abelian sections close to the group size.

Such groups contain a large nilpotent subgroup of class at most 2.

They have many conjugacy classes.

Abstract

Given a finite group $G$ and a generating set $S \subseteq G$, 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 $2$.