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

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.