This question is bit long but please help me:

1.Let $\displaystyle N \triangleleft G$, where $\displaystyle G$ is finite group and $\displaystyle P$ be Sylow subgroup of $\displaystyle G$. Show $\displaystyle P \cap N$ is Sylow p-subgroup of $\displaystyle N$.

Let $\displaystyle |G|=p^ar$ $\displaystyle |N|=p^bs$ , $\displaystyle (r,p)=(s,p)=1$, $\displaystyle a \geq b$. And I think $\displaystyle |PN|= \frac{|P||N|}{|P \cap N|}$ have something to do with this question, but I have no idea.

2.Consider Sylow 2-subgroups $\displaystyle S_3$. Show if $\displaystyle P$ is Sylow p-subgroup of $\displaystyle G$ then $\displaystyle P \cap N$ is not Sylow p-subgroup of $\displaystyle N$ if $\displaystyle N \ntriangleleft G$.

Don't know how to do it at all, please help

Thank you.