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, so
$$
\boxed{H^1(G,\mathbb Q)=0}.
$$
Degree-zero cohomology is the <invariant submodule>; the action is trivial, so
$$
\boxed{H^0(G,\mathbb Q)=\mathbb Q}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-151/4/a/v/solution