Let $G$ be a finite group.
If you consider $H^{n+1}(G,\mathbb{Z})$ and $H_{n}(G,\mathbb{Z})$ (with trivial actions) you will get that these are isomorphic. One can do this by considering instead the cohomology with coefficients in the circle. Indeed, by the universal coefficient theorem and divisibility of $S^1$ $H^n(G,S^1)\cong H_{n}(G,\mathbb{Z})$ and by the long exact sequence in cohomology associated to $\mathbb{Z}\rightarrow\mathbb{R}\rightarrow S^1$ you will get that $H^n(G,S^1)\cong H^{n+1}(G,\mathbb{Z})$. I suppose this is a well known fact. I am not a group cohomologist so I was wondering whether this is correct? Moreover, is there anywhere I can read about these sort of relationships when the group is finite? Also, is there another way of getting this result without using the universal coefficient theorem?