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

被引：0
作者：
Azizollah Azad
Mohammad A. Iranmanesh
Cheryl E. Praeger
Pablo Spiga
机构：
[1] Arak University,Department of Mathematics, Faculty of Sciences
[2] Yazd University,Department of Mathematics
[3] The University of Western Australia,School of Mathematics and Statistics
来源：
Journal of Algebraic Combinatorics | 2011年 / 34卷
关键词：
General linear group; Cyclic matrix; Non-commuting subsets of finite groups; Non-commuting graph;
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摘要：
In this paper we introduce and study a family \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}_{n}(q)$\end{document} of abelian subgroups of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rm GL}_{n}(q)$\end{document} covering every element of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rm GL}_{n}(q)$\end{document}. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}_{n}(q)$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|\mathcal{A}_{n}(q)|$\end{document}. This leads to upper and lower bounds which show in particular that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1q^{-n}\leq \frac{|\mathcal{A}_n(q)|}{|\mathrm{GL}_n(q)|}\leq c_2q^{-n}$$\end{document} for explicit positive constants c1,c2. We also prove that similar upper and lower bounds hold for q=2.
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页码：683 / 710
页数：27
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