Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 4 / 3 / ii / Solution

Modulo $G_3$, the subgroup $G_2=G^p$ is central because $[G_2,G]\subseteq G_3$, and has exponent dividing $p$ because $G_2^p=G_3$. Thus the group $G/G_3$ has nilpotency class at most two. Collection givessince $[y,x]\in G_2$ and $p\mid \left(\genfrac{}{}{0ex}{}{p}{2}\right)$. Also replacing $x$ by $xu$ with $u\in G_2$ leaves its $p$-th power unchanged modulo $G_3$. Hence the assignment is well-defined and is a group homomorphism. Its image contains all the generators $x^p{G}_3$ of $G_2/G_3$, so it is onto: