Abelian coverings of finite general linear groups and an application to their non-commuting graphs
نویسندگان
چکیده
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 ≤ |An(q)| |GLn(q)| ≤ c2q −n for explicit positive constants c1, c2. We also prove that similar upper and lower bounds hold for q = 2. For the 100th anniversary of the birth of B.H. Neumann. The paper forms part of the Australian Research Council Federation Fellowship Project FF0776186 of the third author. The fourth author is supported by UWA as part of the Federation Fellowship project. The second author is supported by Research Council of Yazd University. A. Azad ( ) Department of Mathematics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran e-mail: [email protected] M.A. Iranmanesh Department of Mathematics, Yazd University, Yazd 89195-741, Iran e-mail: [email protected] C.E. Praeger · P. Spiga School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia C.E. Praeger e-mail: [email protected] P. Spiga e-mail: [email protected] 684 J Algebr Comb (2011) 34:683–710 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 An(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.
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