Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol