I have earlier this year in January tried some questions in Group theory but couldn't post them due to my illness.
If $P$ is a normal Sylow $p$-subgroup of a finite group $G$ and $f\colon G\to G$ is an endomorphism, then prove that $f(P) \lt P$.
Let $x, y \in f(P)$.
I have prove that $f(P )$ is a subgroup. But why is it subgroup of $P$ I am unable to prove it? Can you please tell what property of normal p -subgroups should I use?