Computation of the first group cohomology with coefficeint in $\mathbb Z_2$

Question

How can I compute the first cohomology group $H^1(G,\mathbb Z_2G)$, where $G$ is the integer numbers i.e. $\mathbb Z$?

Answer 1

The action of $\mathbf Z$ on $\mathbf Z/2$ is necessarily trivial. Therefore, the group of cocycles is just $\mathrm{Hom}(\mathbf Z,\mathbf Z/2)$. Its subgroup of coboundaries being trivial, one has $$ H^1(\mathbf Z,\mathbf Z/2)=\mathrm{Hom}(\mathbf Z,\mathbf Z/2). $$