The First Group (co)homology of a Group G with Coefficients in Some G–Modules

Abstract

Let G be a group. We give some formulas for the first group homology and cohomology of a group G with coefficients in an arbitrary G–module Ž. More explicit calculations are done in the special cases of free groups, abelian groups and nilpotent groups. We also perform calculations for certain G–module M, by reducing it to the case where the coefficient is a G–module Ž. As a result of the well known equalities H ₁(X,M) = H ₁ (π ₁(X),M) and H ¹(X,M) = H ¹ (π ₁(X),M), for any G–module M, we are able to calculate the first homology and cohomology groups of topological spaces with certain local system of coefficients.