The Gill-Guillot commuting graph for sporadic and related groups

TL;DR

This paper investigates the connectivity properties of the Gill-Guillot commuting graph within specific finite groups, notably quasisimple groups related to sporadic simple groups and some with exceptional Schur multipliers.

Contribution

It characterizes the connectivity of the Gill-Guillot graph for these particular classes of finite groups, extending understanding of their algebraic structure.

Findings

The graph's connectivity varies depending on the group's structure.

Identifies conditions under which the graph is connected or disconnected.

Provides new insights into the interplay between group properties and graph connectivity.

Abstract

Let $G$ be a finite group and $\mathcal{C}$ a normal subset of $G$. The Gill-Guillot graph has vertices $\mathcal C$ and $x, y \in \mathcal C$ are adjacent if and only if $x$ and $y$ commute and $\{xy^{-1},x^{-1}y\} \cap \mathcal C$ is non-empty. We study the connectivity of this graph for quasisimple groups with $G/Z(G)$ a sporadic simple group and for certain simple groups with exceptional Schur multiplier.