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I need help with the following exercise, which, judging by its position in the book, should follow more or less directly from the Sylow theorems.


Let G,G' be two finite groups and $\phi$ a homomorphism of $G$ onto G'. Let $p$ be a prime number and $P,P_1$ two Sylow $p$-subgroups of $G$ such that $\phi(P)=\phi(P_1)$. Show that there exists $x\in\text{Ker}(\phi)$ such that $P=xP_1x^{-1}$.


Well, I know that there exists $y\in G$ such that $P=yP_1y^{-1}$ and it's easy to see that the set of $x$ such that $P=xP_1x^{-1}$ is $yN_G(P_1)$. So we have to show $yN_G(P_1)\cap\text{Ker}(\phi)\neq\emptyset.$

But what now? A hint would be most welcome.

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    Even though it is early in the book, it is still a new, hard, and important idea: use the Sylow theorems in subgroups containing both Sylow subgroups. Jyrki's answer is exactly it, and a very important technique.2011-10-19

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Hint: Recall that if $N\unlhd G$ and $K\le G$, then the set of products $NK=\{nk\mid n\in N, k\in K\}$ is also a subgroup of $G$. The condition \phi(P)=\phi(P') means that the subgroups $(\ker\phi) P$ and (\ker\phi) P' are equal. Let $H$ be that subgroup of $G$. What do the Sylow theorems tell you about the Sylow $p$-subgroups of $H$? Can you list a few of those?