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

In this paper we introduce and study a family An(q) of abelian subgroups of GLn(q) covering every element of GLn(q). We show that An(q) contains all the centralizers of cyclic matrices and equality holds if q > n. For q > 2, we obtain an infinite product expression for a probabilistic generating function for |An(q)|. This leads to upper and lower bounds which show in particular that c1q-n ≤ |A n(q)| |GLn(q)| ≤ c2q-n for explicit positive constants c1, c2. We also prove that similar upper and lower bounds hold for q = 2. A subset X of a finite group G is said to be pairwise non-commuting if xy yx for distinct elements x,y in X. As an application of our results on A n(q), we prove lower and upper bounds for the maximum size of a pairwise non-commuting subset of GLn(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. © 2011 Springer Science+Business Media, LLC.