Exercise 25, § 6

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