Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 5 / a

Choose a normalized set-theoretic section $s:H\to E$ of $\pi$, so $s(1)=1$. Define the extension cocycleAssociativity of $s(h)s(k)s(l)$ gives, in multiplicative notation for $M$,which is exactly the two-cocycle identity. Thus $\varphi$ represents the class of the group extension in $H^2(H,M)$.

The fiber product of groupsis a group under componentwise multiplication. The maps $m\mapsto (m,1)$ and $(e,g)\mapsto g$ give an exact sequenceThe section $g\mapsto (s(f(g)),g)$ has extension cocycleTherefore this pullback extension represents $f^{\ast }([\varphi ])\in H^2(G,M)$.

Assume $f$ is injective. If two elements $(e,g),(e^{\prime },g^{\prime })\in E^{\prime }$ have the same image under the first projection $E^{\prime }\to E$, then $e=e^{\prime }$. Applying $\pi$ gives $f(g)=f(g^{\prime })$, so injectivity of $f$ gives $g=g^{\prime }$. Hence the natural map $E^{\prime }\to E$ is injective.