Abelian coverings of finite general linear groups and an application to their non-commuting graphs期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

Email author" target="_blank">Azizollah?Azad Email author Mohammad?A?Iranmanesh Cheryl?E?Praeger Pablo?Spiga

Institution:

(1) AT & T Labs—Research, Florham Park, USA;(2) School of Engineering and Applied Sciences, Harvard University, Cambridge, USA

Abstract:

In this paper we introduce and study a family A_n(q)\mathcal{A}_{n}(q) of abelian subgroups of GL_n(q){\rm GL}_{n}(q) covering every element of GL_n(q){\rm GL}_{n}(q). We show that A_n(q)\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 |A_n(q)||\mathcal{A}_{n}(q)|. This leads to upper and lower bounds which show in particular that

c₁q^-n ￡ \frac\|A_n(q)\|\|GL_n(q)\| ￡ c₂q^-nc_1q^{-n}\leq \frac{\|\mathcal{A}_n(q)\|}{\|\mathrm{GL}_n(q)\|}\leq c_2q^{-n}

Keywords:
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