Commuting involution graphs of linear groups

TL;DR

This paper determines the diameters of commuting involution graphs in linear groups over various fields, providing new insights into their structure and extending previous results to special cases like projective groups and characteristic 2 fields.

Contribution

It uniquely computes the diameters of commuting involution graphs for special and general linear groups over arbitrary fields, including cases over finite fields and characteristic 2.

Findings

Diameters of commuting involution graphs for special and general linear groups over arbitrary fields.

Diameter results for projective special linear groups over finite fields.

Complete classification of linear groups over finite fields based on involution distances.

Abstract

In this paper, we determine the diameter of the commuting involution graphs of special and general linear groups over an arbitrary field. It turns out that our results also determine the diameter for certain projective special linear groups over finite fields. Moreover, we find the diameter of the commuting graphs of general linear groups on the set of all involutions over a field of characteristic 2, which completes the diameter of general linear groups on the set of all involutions. As an application, we classify the structure of the four-dimensional linear groups over finite fields according to the distance from a fixed involution.