Is there a general simple theorem for the third cohomology of cyclic groups $H\_3(\mathbb{Z}\_n, U(1))= ?$. In particular, I am interested in finding $H\_3(\mathbb{Z}\_8, U(1))$. I know the answer can be found using GAP, but I wanted a formal theorem, and also I don't have access to the HAP package of GAP, which is used for such computations. 



Third (co-) homology of Cyclic groups
For cyclic groups, the modified cohomology groups are periodic with period 2. In particular, there are isomorphisms $$H\_3(\mathbb Z/n\mathbb Z,U(1))=\hat H^{-4}(\mathbb Z/n\mathbb Z,U(1))\cong \hat H^0(\mathbb Z/n\mathbb Z,U(1))=U(1)^{\mathbb Z/n\mathbb Z}/N(U(1))=U(1)/n=0$$


Here $N$ is the norm map $x\mapsto \sum\_{a\in \mathbb Z/n\mathbb Z} a\cdot x$.


