  In this paper, we consider for any free presentation $G = F/R$ of a group $G$
the coinvariance $H_{0}(G,R_{ab}^{\otimes n})$ of the $n$-th tensor power of
the relation module $R_{ab}$ and show that the homology group
$H_{2n}(G,{\mathbb Z})$ may be identified with the limit of the groups
$H_{0}(G,R_{ab}^{\otimes n})$, where the limit is taken over the category of
these presentations of $G$. We also consider the free Lie ring generated by the
relation module $R_{ab}$, in order to relate the limit of the groups
$\gamma_{n}R/[\gamma_{n}R,F]$ to the $n$-torsion subgroup of $H_{2n}(G,{\mathbb
Z})$.
