Homological methods in group varieties

Abstract

In papers by Stallings [9] and the author [10], [11] the homology theory of groups has been applied to central series, and meaningful group theoretical results have been obtained. In the present paper we show how similar homological methods can be used to obtain interesting results in arbitrary varieties of groups. It is not surprising that the methods are most successful if applied to problems involving central series. It is the nature of a paper introducing new methods to repeat many well known results. However the approach presented here leads to an interestingly unifying point of view. Also, it is possible to simplify the proofs of many well known results. Given a variety ~ , we first define a functor S o from ~ to abelian groups. This functor is defined in terms of the integral second homology group functor H 2 ( , Z). We then prove that a surjective group homomorphism G ~ Q in ~ , with kernel N, gives rise to an exact sequence