Minimum polynomials of the elements of prime order in representations of quasi-simple groups.
type: Item
instance of: scholarly article
title: Minimum polynomials of the elements of prime order in representations of quasi-simple groups.
zbMATH Open document ID: 1163.20004
published in: Journal of Algebra
publication date: 2008-11-12T00:00:00Z
review text: Let \(p\) be a prime number, \(G\leq\text{GL}(n,\mathbb{C})\). We call \(G\) an \(N_p\)-group if it contains a matrix \(g\) such that \(g^p\) is a scalar matrix, \(g\) has at most \(p-1\) distinct eigenvalues and \(g\) does not belong to a proper normal subgroup of \(G\). As no \(N_2\)-group exists for \(n>1\) we assume \(p>2\). In this paper quasi-simple groups of \(N_p\)-type are determined. The cases where \(g\) has at most \(p-2\) distinct eigenvalues or a Sylow \(p\)-subgroup of \(G\) is cyclic are considered by the author earlier (Theorems 2.1 and 2.2). The main result considers the case where \(g\) has exactly \(p-1\) distinct eigenvalues and a Sylow \(p\)-subgroup of \(G\) is noncyclic (Theorem 1.1).
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5365177
zbMATH Keywords: finite linear groups
zbMATH Keywords: representations of finite groups
zbMATH Keywords: quasi-simple groups
author: A. Zalesski
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.02.037
OpenAlex ID: W2031461658
cites work: Generic Sylow theorems for the reductive groups over finite fields
cites work: Q4217740
cites work: Q3684278
cites work: The 2-Transitive Permutation Representations of the Finite Chevalley Groups
cites work: Character Table and Blocks of Finite Simple Triality Groups 3 D 4 (q)
cites work: Q3846428
cites work: Q3955560
cites work: Linear groups over \(\text{GF}(2^k)\) generated by a conjugacy class of a fixed point free element of order 3
cites work: Q4438878
cites work: Weil representations associated to finite fields
cites work: Cross-characteristic character and fixed point space ratios for groups of Lie type
cites work: The local structure of finite groups of characteristic 2 type
cites work: Q4375273
cites work: Singer-Zyklen in klassischen Gruppen
cites work: Q3934684
cites work: On the minimal degrees of projective representations of the finite Chevalley groups
cites work: Centralizers of Semisimple Elements in Finite Twisted Groups of Lie Type
cites work: Subgroups of Maximal Rank in Finite Exceptional Groups of Lie Type
cites work: Minimal characters of the finite classical groups
cites work: Some generalizations of the Weil representations of the symplectic and unitary groups
cites work: Sylow p-Subgroups of the Classical Groups Over Finite Fields with Characteristic Prime to p
cites work: Q3475470
cites work: Q3810006
cites work: The eigenvalue 1 of matrices of complex representations of finite Chevalley groups
cites work: Minimal Polynomials and Eigenvalues of p -Elements in Representations of Quasi-Simple Groups with a Cyclic Sylow p -Subgroup
cites work: Q3125647
cites work: THE NUMBER OF DISTINCT EIGENVALUES OF ELEMENTS IN FINITE LINEAR GROUPS
DOI: 10.1016/J.JALGEBRA.2008.02.037
description: scientific article; zbMATH DE number 5365177