 Integral cohomology of a finite cyclic group

For the trivial group action of the finite cyclic group $C_m$ on the integers,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.