Linear transvection groups and embedded polar spaces
Let \(K\) be a skew field and \(V\) a vector space over \(K\). Assume that \(G\) is a subgroup of \(\text{GL}(V)\) such that (1) \(G\) is generated by a conjugacy class \(\Sigma\) of Abelian subgroups of \(G\); (2) for \(A,B\in\Sigma\), either \([A,B]=1\) or \(\langle A,B\rangle\) is a rank 1 group (i.e., \(A\neq B\), and for each \(a\in A^\#\), there exists some \(b\in B^\#\) with \(A^b=B^a\), and vice versa). Then \(\Sigma\) is a class of abstract transvection groups [see \textit{F. G. Timmesfeld}, Adv. Math. 142, No. 1, 1-150 (1999)]. Observe that it is not assumed that for all \(A\) in \(\Sigma\) the commutator \([V,A]\) is \(1\)-dimensional nor that \(C_V(A)\) is a hyperplane of \(V\). The main result obtained by the authors is a common characterization of the various classical groups different from the special linear group, as groups containing a class \(\Sigma\) of abstract transvection groups satisfying some additional conditions. The characterized groups are isometry groups arising from hermitian, antihermitian, quadratic, or pseudo-quadratic forms. The class of abstract transvection groups is here a class of groups of isotropic transvections.
scientific article; zbMATH DE number 1336952