Subgroups of classical groups generated by transvections or Siegel transvections. I: Embeddings in linear groups
Transvections and Siegel transvections play an important role in the theory of classical groups. Let \(Y\) be a finite-dimensional classical group and \(G\) a subgroup of \(Y\) generated by (Siegel) transvections. Assume \(G\) is defined over a field and has rank at least 3. Under some weak restrictions in characteristic 2, the author proves that \(G/Z(G)\) is isomorphic to a finite-dimensional projective classical group \(S/Z(S)\). In this situation the classical group \(S\) is said to embed in the classical group \(Y\) such that (Siegel) transvections act as (Siegel) transvections. The author gives all the possible such embeddings in linear groups by using methods of geometric algebra and the fundamental theorem of projective geometry.
finite-dimensional projective classical groups
fundamental theorem of projective geometry
scientific article; zbMATH DE number 1121162