Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 4 / a

The inhomogeneous group cochain group iswith $C^0(G,M)=M$. Its group coboundary $d:C^r(G,M)\to C^{r+1}(G,M)$ isOne checks that $d^2=0$, and group cohomology is $H^r(G,M)=\ker d/\mathrm{im}d$.

A crossed homomorphism is a map $\varphi :G\to M$ satisfyingFor a one-cochain, the formula in part i givesso the crossed homomorphisms are exactly the one-cocycles. A zero-cochain $m\in M$ has coboundary $g\mapsto gm-m$, the principal crossed homomorphism associated with $m$. Therefore

For $g,h\in G$, substitute the definition of $\psi$ and in the first sum replace $\gamma$ by $\delta g$:The two-cocycle identity at $(\delta ,g,h)$ says that the expression in parentheses is $\delta \varphi (g,h)$. Every summand is therefore $\varphi (g,h)$, and

Let $\varphi \in Z^2(G,\mathbb{Q})$. Part iii gives $d\psi =|G|\varphi$. Since division by $|G|$ is possible in the rational numbers,is a two-coboundary. HenceThis is a degree-two instance of the vanishing of finite-group cohomology when the group order is invertible on the coefficient module.

For the trivial action, crossed homomorphisms $G\to \mathbb{Q}$ are ordinary group homomorphisms, while principal crossed homomorphisms vanish. The image of a finite group in the torsion-free additive group $\mathbb{Q}$ must be trivial, soDegree-zero cohomology is the invariant submodule; the action is trivial, so