I'm trying to teach myself group cohomology and I'm trying to compute a simple example, $H^{n}(\mathbb{Z} / \langle 2 \rangle , \mathbb{Z})$.
I know that I form the chain complex as $C^\cdot: ...\rightarrow Hom_{\mathbb{Z[\mathbb{Z} \langle 2 \rangle]}}((\mathbb{Z}/ \langle 2 \rangle)^n,\mathbb{Z})\xrightarrow{d ^n}Hom_{\mathbb{Z[\mathbb{Z} \langle 2 \rangle]}}((\mathbb{Z}/ \langle 2 \rangle)^{n-1},\mathbb{Z})\xrightarrow{d^{n-1}}... $ where $d^n$ is defined as $d^n(\psi)(x_1,...,x_n)=x_1\psi(x_2,...,x_n) +\sum_{i=2}^{n-1}(-1)^ix_i\psi(x_1,...,\bar x_i,...,x_n)+x_n\psi(x_1,...,x_{n-1})$ and the cohomology is just the homology of this complex. Can someone work through an example of find the kernels and images of these maps?I've stared at the definitions for a while and can't really figure out where to start.