Split chief factors and cohomology
Let G be a finite group. Let \(k=F_ p\) be the field with p elements. A simple kG-module M is called a split chief factor if there exist normal subgroups N and K of G such that K/N is isomorphic to M and such that G/N splits over K/N. For any given chief series of G, the number s(G,M) of split chief factors in this series is known to be independent of the choice of the series. The principal result generalizes a result of Gaschütz: Theorem. \(s(G,M)=\dim_ DH^ 1(G,M)-\dim_ DH^ 1(G/CM,M)\), where \(D=End_{kG}M\) and \(CM=\ker nel\) of the action of G on M. In conjunction of an older result of the author, one gets a converse of a result of Gaschütz: Corollary. G is p-solvable if and only if \(s(G,M)=\dim_ DH^ 1(G,M)\) for every simple kG-module M. These results have equivalent formulations in terms of modular representation theory.
scientific article; zbMATH DE number 3995976
scientific article