Abstract
In this paper we introduce and study a family \(\mathcal{A}_{n}(q)\) of abelian subgroups of \({\rm GL}_{n}(q)\) covering every element of \({\rm GL}_{n}(q)\). We show that \(\mathcal{A}_{n}(q)\) 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 \(|\mathcal{A}_{n}(q)|\). This leads to upper and lower bounds which show in particular that
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 \(xy\not=yx\) 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.
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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.
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Azad, A., Iranmanesh, M.A., Praeger, C.E. et al. Abelian coverings of finite general linear groups and an application to their non-commuting graphs. J Algebr Comb 34, 683–710 (2011). https://doi.org/10.1007/s10801-011-0288-2
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DOI: https://doi.org/10.1007/s10801-011-0288-2