On the diameter of Cayley graphs of classical groups with generating sets containing a transvection

TL;DR

This paper proves a logarithmic diameter bound for Cayley graphs of certain classical groups when the generating set includes a transvection, supporting Babai's conjecture in these cases.

Contribution

It establishes a diameter bound for Cayley graphs of $PSL(n,q)$, $PSp(n,q)$, and $PSU(n,q)$ with generating sets containing a transvection, under specific conditions.

Findings

Diameter of Cayley graphs is bounded by a logarithmic function of group size.

Supports Babai's conjecture for classical groups with transvection-containing generating sets.

Provides bounds for groups with odd q, excluding q=9 or 81.

Abstract

A well-known conjecture of Babai states that if $G$ is any finite simple group and $X$ is a generating set for $G$, then the diameter of the Cayley graph $Cay(G,X)$ is bounded by $\log|G|^c$ for some universal constant $c$. In this paper, we prove such a bound for $Cay(G,X)$ for $G=PSL(n,q),PSp(n,q)$ or $PSU(n,q)$ where $q$ is odd, under the assumptions that $X$ contains a transvection and $q\neq 9$ or $81$.