A <crossed homomorphism> is a map $\phi:G\to M$ satisfying
$$
\phi(gh)=\phi(g)+g\phi(h).
$$
For a one-cochain, the formula in part i gives
$$
(d\phi)(g,h)=g\phi(h)-\phi(gh)+\phi(g),
$$
so 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
$$
\boxed{H^1(G,M)=\frac{\{\text{crossed homomorphisms }G\to M\}}
{\{g\mapsto gm-m:m\in M\}}}.
$$

Solution

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

For $m\in M$, the map $g\mapsto g\cdot m-m$ is a principal crossed homomorphism. It is the <group coboundary> of the zero-cochain $m$, so first <group cohomology> is the quotient of crossed homomorphisms by principal crossed homomorphisms.

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 Incoming links: Principal crossed homomorphism