= Integral cohomology of a finite cyclic group
For the trivial <group action> of the <finite cyclic group> $C_m$ on the <integers>,
$$
H^0(C_m,\mathbb Z)=\mathbb Z,\qquad H^{2k+1}(C_m,\mathbb Z)=0\ (k\geq0),\qquad H^{2k}(C_m,\mathbb Z)=\mathbb Z/m\mathbb Z\ (k\geq1).
$$
The <periodic resolution of a finite cyclic group> alternates multiplication by $t-1$ and the norm $N=1+t+\cdots+t^{m-1}$. Applying the <Hom functor> with trivial coefficients makes those maps zero and multiplication by $m$, respectively. Their <kernels> and <images> give the displayed <group cohomology>. In particular $H^1=0$, while $H^2$ detects the cyclic central extensions appearing in <second group cohomology classifies group extensions>.
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