Sylow p-subgroups

Quote:

Originally Posted by dori1123

Let $G$ be a finite group and $p$ be a prime.
Prove that if $Q \in Syl_p(G)$ and $H$ is a subgroup of $G$ containing $Q$ then $Q \in Syl_p(H)$ .
$Syl_p(G)$ is the set of Sylow p-subgroups of G.

Write $|G|=p^a\cdot n$ where $p\not | n$ . Therefore $|Q| = p^a$ . Since $H\subseteq G$ it means $|H| = p^b \cdot m$ where $0\leq b\leq a$ and $m|n$ . But $Q\subseteq H$ so $|Q|$ divides $|H|$ , thus, $b=a$ . We see that $Q$ is a Sylow subgroup because it has the largest exponent of $p$ dividing $|H|$ .