# Computation of group cohomology $H^d(\mathbb {Z},\mathbb {Z}_2 )$, $H^d(\mathbb {Z},\mathbb {Z}_2 \times \mathbb {Z}_2)$

Is there some way to compute group cohomology $H^d(\mathbb {Z},\mathbb {Z}_2 )$ , $H^d(\mathbb {Z},\mathbb {Z}_2 \times \mathbb {Z}_2)$ and $H^d(\mathbb {Z}\times \mathbb {Z},\mathbb {Z}_2 \times \mathbb {Z}_2)$? I'm not a math student and I only know the definition of group cohomology. But only from the definition, I can't solve these two questions.

My question:

1. How to solve these two examples? I want to learn group cohomology from seeing calculation some examples. You can tell me which tricks or formula I need to use. Or you can just refer me some reference where I can find the how to solve it. If it's too hard to solve general $d$, how to solve for $d=1,2,3$ which I'm interested in.

1. Assuming that the actions of $\mathbb{Z}$ are trivial in all cases, the universal coefficient theorem still holds by abstract nonsense. Furthermore, $H^*(\mathbb{Z}, \mathbb{Z}) = H^*(K(\mathbb{Z}, 1), \mathbb{Z}) = H^*(S^1, \mathbb{Z})$; the usual proof is to show that the cochains $C^*(\pi)$ in singular cohomology produce an appropriate projective resolution. (Since it sounds like from the post that you're interested in group cohomology but just getting started in it, I'll also mention that going through calculations is not the right way to learn group cohomology. (Or much of algebraic topology or this kind of group theory, for that matter.) A direct computation is seldom enlightening even in the few cases where it's possible. Besides, the goal of group cohomology is not to compute the cohomology of groups explicitly, any more than the goal of the study of differential equations is to obtain explicit solutions of differential equations.)