Conjugacy classes in Sylow \(p\)-subgroups of \(GL(n,q)\)
type: Item
instance of: scholarly article
title: Conjugacy classes in Sylow \(p\)-subgroups of \(GL(n,q)\)
zbMATH Open document ID: 0777.20015
DOI: 10.1016/0021-8693(92)90085-Z
author: A. Vera-López
author: J. M. Arregi
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: Let \(UT_ n(q)\) be the group of upper unitriangular matrices of degree \(n\) over the field of \(q\) elements where \(q\) is a power of a prime \(p\), and \(r_ n(q)\) the number of conjugacy classes of elements of \(UT_ n(q)\). It is proved that, for \(n\geq 6\), the following inequality holds: \(r_ n(q)\leq q^{n(n+1)/6}(n-1)!2^{n-1}\) (Theorem 5.4). This upper bound for \(r_ n(q)\) improves those given by \textit{G. Higman} [Proc. Lond. Math. Soc., III. Ser. 10, 24-30 (1960; Zbl 0093.02603)] in Theorems 3.2-3.5. Set \(q=p^ t\) and \(tn(n-1)/2=2m+e\), with \(e\in\{0,1\}\). Then there exists a non-negative number \(k\) such that \[ r_ n(q)=m(p^ 2-1)+p^ e+{{(p-1)q(q^{n-1}-1)} \over {p(q-1)}} -h(n,q)+k(p^ 2-1)(p- 1), \] where \(h(n,q)=(n-1)(p^ 2-p)\) or \(((n-2)/2)(p^ 2-1)+ (p-1)\) or \(((n-1)/2)(p^ 2-1)\) according as \(t\) is an even number, \(n\) is an even number and \(t\) an odd number, or both \(n\) and \(t\) are odd numbers, respectively (Theorem 7.1).
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 130512
zbMATH Keywords: group of upper unitriangular matrices
zbMATH Keywords: number of conjugacy classes
reviewed by: Ju. I. Merzljakov
MaRDI profile type: Publication
cites work: Enumerating p -Groups. I: Inequalities
cites work: Q5517708
cites work: Enumerating p -Groups
cites work: On the number of conjugacy classes in a finite p-group
cites work: On the number of conjugacy classes in a finite group
description: scientific article; zbMATH DE number 130512