Maximal Cocliques in $\operatorname{PSL}_2(q)$

TL;DR

This paper studies the structure of the generating graph of the group PSL_2(q), classifies large cocliques for prime q, and explores geometric examples that challenge these classifications when q is a prime power.

Contribution

It provides a classification of large cocliques in PSL_2(q) for prime q and highlights the limitations of existing methods for prime powers.

Findings

Classified large cocliques in PSL_2(q) for prime q

Presented a geometric example contradicting the prime case results for prime powers

Showed methods for prime q do not extend straightforwardly to prime powers

Abstract

The generating graph of a finite group is a structure which can be used to encode certain information about the group. It was introduced by Liebeck and Shalev and has been further investigated by Lucchini, Mar\'oti, Roney-Dougal and others. We investigate maximal cocliques (totally disconnected induced subgraphs of the generating graph) in $\operatorname{PSL}_2(q)$ for $q$ a prime power and provide a classification of the `large' cocliques when $q$ is prime. We then provide an interesting geometric example which contradicts this result when $q$ is not prime and illustrate why the methods used for the prime case do not immediately extend to the prime-power case with the same result.