On the construction of matrix representations and their extensions
type: Item
instance of: scholarly article
title: On the construction of matrix representations and their extensions
zbMATH Open document ID: 0729.20003
DOI: 10.1016/0019-3577(90)90023-G
author: Eric B. Kuisch
published in: Indagationes Mathematicae. New Series
publication date: 1990-01-01T00:00:00Z
review text: Let \(G=E\cdot H\) be the semi-direct product of finite groups E and H, (E is normal in G), with the following conditions: (1) E is a nonabelian p- group, p odd, and its center Z(E) is cyclic; (2) E/Z(E) is elementary abelian; (3) Z(E)\(\subseteq Z(E\cdot H)\); (4) H is a complement of Inn(E) in \(Aut_{Z(E)}(E)\). The main result of this paper is the following: If \(\psi\) is a faithful representation of E then an extension of \(\psi\) to G can be constructed. The author uses many steps to get to this aim. Some of these are studying the structure of E, deriving faithful representations of E, constructing a complement H of Inn(E) in \(Aut_{Z(E)}(A)\), constructing projective representations of \(E\cdot H\), and finally generalizing the results to arbitrary complements. In the last section of this paper, the author applies these results to construct certain characters of Sp(2m,q), \(m\geq 1\) and q an odd prime power.
reviewed by: B.G.Basmaji
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 4204639
zbMATH Keywords: semi-direct product
zbMATH Keywords: finite groups
zbMATH Keywords: extension
zbMATH Keywords: faithful representations
zbMATH Keywords: complement
zbMATH Keywords: projective representations
zbMATH Keywords: characters
MaRDI profile type: Publication
cites work: Characters of groups with normal extra-special subgroups
cites work: Automorphisms of extra special groups and nonvanishing degree 2 cohomology
cites work: On the Clifford collineation, transform and similarity groups. I.
cites work: Q4103728
cites work: Characters of Solvable and Symplectic Groups
cites work: Some Representations of Classical Groups
cites work: Representations of symplectic groups
cites work: Endliche Gruppen I
cites work: Q5675788
cites work: Weil representations associated to finite fields
description: scientific article; zbMATH DE number 4204639