Sharply transitive linear algebraic groups
The author studies Zariski-closed linear groups \(G \leq GL_ n k\) which act sharply transitively on the non-zero vectors of \(k^ n\), where \(k\) is a field of characteristic zero. If \(n\) is square-free or if the cohomological dimension of \(k\) is at most 1, then he proves that \(G\) is the multiplicative group of a field extension of degree \(n\) over \(k\). If \(n \leq 15\), then \(G\) is contained in a semi-linear group \(\Gamma L_ 1D\) for some skew field \(D\) of dimension \(n\) over \(k\). These results are proved by passing to the Lie algebra \(L\) of \(G\) and by investigating the representation of \(L\) in \(k^ n\). The author's arguments simplify parts of a related paper by the reviewer [Forum Math. 1, 81-101 (1989; Zbl 0649.20045)]. The results on \(G\) (or \(L\)) can be restated in terms of nearfields (or of left-symmetric division algebras).
scientific article; zbMATH DE number 166879
scientific article