Abelian coverings of finite general linear groups and an application to their non-commuting graph

TL;DR

This paper constructs a family of abelian subgroups covering all elements of al_n(q) and applies this to determine bounds for the size of the largest pairwise non-commuting subset in al_n(q), with explicit formulas when q>n.

Contribution

It introduces a new family of abelian subgroups covering al_n(q) and derives bounds and explicit formulas for non-commuting sets in al_n(q).

Findings

al_n(q) can be covered by a specific family of abelian subgroups.

Explicit formula for the size of non-commuting sets when q>n.

Upper bounds for non-commuting sets when q=2.

Abstract

In this paper we introduce and study a family $\mathcal{A}_n(q)$ of abelian subgroups of $\GL_n(q)$ covering every element of $\GL_n(q)$. We show that $\mathcal{A}_n(q)$ contains all the centralisers of cyclic matrices and equality holds if $q>n$. Also, for $q>2$, we prove a simple closed formula for the size of $\mathcal{A}_n(q)$ and give an upper bound if $q=2$. A subset $X$ of a finite group $G$ is said to be pairwise non-commuting if $xy\not=yx$, for distinct elements $x, y$ in $X$. As an application of our results on $\mathcal{A}_n(q)$, we prove lower and upper bounds for the maximum size of a pairwise non-commuting subset of $\GL_n(q)$. (This is the clique number of the non-commuting graph.) Moreover, in the case where $q>n$, we give an explicit formula for the maximum size of a pairwise non-commuting set.