Group homology ($H_n(G,M)$)

Group homology is obtained by tensoring a projective resolution of the trivial $\mathbb{Z}G$-module with a coefficient module and taking homology.

The Schur multiplier of a group $G$ is its second integral group homology group:

For a free presentation $G\cong F/R$, Hopf's formula is

For an abelian group $A$,For a finite direct sum of cyclic groups this gives one summand $C_{\gcd (m,n)}$ for every pair $C_m,C_n$.