Year: 2021
Type: preprint
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 $2$.
Source: arXiv (Cornell University)
Author: Sabatini, Luca
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3176257382