In this paper we introduce and study a family n(q) of abelian subgroups of GLn(q) covering every element of GLn(q). We show that n(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 n(q) . This leads to upper and lower bounds which show in particular that c1q−nn(q)GLn(q)c2q−n for explicit positive constants c 1,c 2. 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 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.
