# Abelian coverings of finite general linear groups and an application to their non-commuting graph - Mathematics > Group Theory

Abelian coverings of finite general linear groups and an application to their non-commuting graph - Mathematics > Group Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: In this paper we introduce and study a family $\mathcal{A} nq$ of abeliansubgroups of $\GL nq$ covering every element of $\GL nq$. We show that$\mathcal{A} nq$ contains all the centralisers of cyclic matrices andequality holds if $q>n$. Also, for $q>2$, we prove a simple closed formula forthe size of $\mathcal{A} nq$ and give an upper bound if $q=2$. A subset $X$of a finite group $G$ is said to be pairwise non-commuting if $xy ot=yx$, fordistinct elements $x, y$ in $X$. As an application of our results on$\mathcal{A} nq$, we prove lower and upper bounds for the maximum size of apairwise non-commuting subset of $\GL nq$. This is the clique number of thenon-commuting graph. Moreover, in the case where $q>n$, we give an explicitformula for the maximum size of a pairwise non-commuting set.

Author: ** A. Azad, M. A. Iranmanesh, C. E. Praeger, P. Spiga**

Source: https://arxiv.org/