Transitivitätsfragen bei Gruppen von Kollineationen. (Transitivity questions for collineation groups) (Q1081682)

The author studies groups of semilinear transformations of a vector space V which are transitive on the set \(T_ k(V)\) of all k-tuples of one- dimensional subspaces generating a subspace of dimension k. He shows: If G is a subgroup of \(\Gamma\) L(V) and \(N\not\subseteq Z(GL(V))\) is a normal subgroup of G, if furthermore, for some \(2\leq k\leq n=\dim V\), G is transitive on \(T_ k(V)\), then N is transitive on \(T_{k-1}(V)\) (Satz 1). This allows the following generalization of a theorem by \textit{H. Völklein} [J. Algebra 78, 341-356 (1982; Zbl 0497.20033)]: If G is a subgroup of \(\Gamma\) L(V) which is transitive on \(T_ k(V)\) and Z(GL(V))\(\not\supseteq GL(V)\cap G\), then (a) SL(V)\(\subset G\) if dim V\(=n\geq 4\) and \(n<2k\leq 2n\), (b) SL(V)\(\subset G\) also for \(2k=n\) if \(n\geq 5\) and the field of scalars is an algebraic extension of a finite field or is algebraically closed.