Maximal cocliques in PSL₂(q)

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, Maróti, Roney-Dougal, and others. We investigate maximal cocliques (totally disconnected induced subgraphs of the generating graph) in ${\text{PSL}}_2(q)$ 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.

Keywords:

• Coclique
• generating graph
• orthogonal group
• projective special linear group

MSC Classification 2010:

• 20E34
• 20D06

Acknowledgments

This work was done as part of my PhD, thus I would like to thank my supervisor Corneliu Hoffman for everything he has done for me thus far.