Cohomology in abstract unit groups

Abstract

A. H. Clifford and S. MacLane [2] considered in 1941 the group of factor-sets H2(F, U) of a finite group F over its abstract unit group U. They proved the main theorem to the effect that H2(F, U) is isomorphic to the multiplicator M of F defined by I. Schur and also several other theorems under the assumption that F is a solvable group. They conjectured that these should hold for general finite groups r. In 1942 A. Weil proved the main theorem for general finite groups F, but this result was not published.' In this short note we shall prove that all the theorems in [2] are valid for general finite groups r, and also we shall extend their results for all (positive, zero, and negative) dimensional cohomology groups.2 1. We shall first prove a general lemma on cohomology groups. Let A be a finite group, and let E be a A-module. Suppose that Al, A2 are two A-submodules which are disjoint: AflrA2=0. Then we have the following commutative diagram such that each row and each column are exact: