Exercise 25, § 6

Let $\phi : G \to G'$ be a surjective homomorphism of finite groups; let $P$ be a Sylow $p$-subgroup of $G$. (a) Let $P_1$ be a Sylow $p$-subgroup of $G$ such that $\phi(P) = \phi(P_1)$. Show that there exists an element $x$ of the kernel of $\phi$ such that $P_1 = xPx^{-1}$. (b) Let $N$ (resp. $N'$) be the normalizer in $G$ (resp. $G'$) of $P$ (resp. $\phi(P)$). Show that $\phi(N) = N'$. In particular, if the order of $G'$ is not divisible by $p$, then $\phi(P) = \{e\}$ and $\phi(N) = G'$.