Projective-resolution definition of group cohomology

Choose a projective resolution $P_{\bullet }\to \mathbb{Z}$ of the trivial $\mathbb{Z}G$-module. For a $\mathbb{Z}G$-module $M$, group cohomology isDifferent projective resolutions give naturally isomorphic groups.

For $\mathbb{Z}G$-modules $M_1,M_2$,because the Hom functor into a finite direct sum splits degree by degree and taking cohomology preserves that splitting.

For $G=⟨x,y\mid xy=yx⟩$, the trivial $\mathbb{Z}G$-module has the free resolution

If $G\cong {\mathbb{Z}}^2$, $I$ is the augmentation ideal, and $M=\mathbb{Z}G/I^2$, thenThe quotient map $M\to \mathbb{Z}G/I\cong \mathbb{Z}$ induces an isomorphism in second cohomology.