1. ## Sylow p-subgroups

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.

2. 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 $|H|$.