Quadratic pairs without components
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.
scientific article; zbMATH DE number 1868096