Quadratic pairs without components
type: Item
instance of: scholarly article
title: Quadratic pairs without components
zbMATH Open document ID: 1024.20003
DOI: 10.1016/S0021-8693(02)00637-3
author: Andrew Chermak
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: A quadratic pair \((G,V)\) is a group \(G\) together with a faithful \(\text{GF}(p)\)-module \(V\) such that \(G=\langle A^G\rangle\) for some elementary Abelian subgroup \(A\) of \(G\) with \([V,A,A]=0\). If \(p>3\) a classical result due to J. Thompson yields that \(F^*(G)\) is a quasisimple group of Lie type. For \(p=3\) there are partial results due to \textit{C.-Y. Ho} [Bull. Inst. Math., Acad. Sin. 1, 155-180 (1973; Zbl 0292.20024), J. Algebra 43, 338-358 (1976; Zbl 0385.20006)] where he assumed \(F^*(G)\) to be quasisimple. In the paper under review the author treats the case that \(F^*(G)=F(G)\). He proves that \(F(G)=O_2(G)\) is a central product of a group of order 2 or 4 by an extraspecial group, further \(G/F(G)\cong A_{2n+1}\), \(A_{2n+2}\), \(\text{GU}(n,2)\), \(\Omega^\varepsilon_{2n}(2)\) or \(\text{Sp}(2n,2)\) with natural action on \(F(G)/Z(F(G))\). This result is a corollary of a result on modules with forms proved by the author in this paper. He considers a \(\text{GF}(2)\)-vector space \(V\) with a nondegenerated alternating bilinear form: Let \(G^*\) be the isometry group of \(V\) and let \(d\in G^*\) such that \([V,d]\) is a nondegenerate subspace of dimension 2 and \(G=\langle d^G\rangle\) be a subgroup of \(G^*\) with \([V,G]=V\). Then he proves that \(G\) and \(V\) are some of the groups and modules occurring in the theorem above. The proof of the theorem is elementary in the sense that it just uses facts from geometric algebra. Reviewer's remark: The author claims that \textit{M. Aschbacher}'s book [Finite group theory, Cambridge University Press (1986; Zbl 0583.20001)] is the only book in English containing some material on geometric algebra. There is at least one further due to \textit{D. E. Taylor} [The geometry of the classical groups, Heldermann (1992; Zbl 0767.20001)] and of course there is Dieudonné's classical book.
reviewed by: Gernot Stroth
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1868096
zbMATH Keywords: faithful modules
zbMATH Keywords: quadratic modules
zbMATH Keywords: quadratic pairs
MaRDI profile type: Publication
cites work: A characterization of Chevalley groups over fields of odd order. I, II
cites work: Q3708174
cites work: Groups generated by a class of elements of order 3
cites work: Q5577154
cites work: Q4043230
cites work: On the quadratic pairs
cites work: Another look at Thompson's quadratic pairs
cites work: Einfache Gruppen, die von einer Konjugiertenklasse von Elementen der Ordnung drei erzeugt werden
cites work: Q3712472
cites work: Abstract root subgroups and quadratic action. With an Appendix by A. E. Zalesskii
description: scientific article; zbMATH DE number 1868096