1. ## Sylow p-subgroups

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

2. Originally Posted by dori1123
Let $\displaystyle G$ be a finite group and $\displaystyle p$ be a prime.
Prove that if $\displaystyle Q \in Syl_p(G)$ and $\displaystyle H$ is a subgroup of $\displaystyle G$ containing $\displaystyle Q$ then $\displaystyle Q \in Syl_p(H)$.
$\displaystyle Syl_p(G)$ is the set of Sylow p-subgroups of G.
Write $\displaystyle |G|=p^a\cdot n$ where $\displaystyle p\not | n$. Therefore $\displaystyle |Q| = p^a$. Since $\displaystyle H\subseteq G$ it means $\displaystyle |H| = p^b \cdot m$ where $\displaystyle 0\leq b\leq a$ and $\displaystyle m|n$. But $\displaystyle Q\subseteq H$ so $\displaystyle |Q|$ divides $\displaystyle |H|$, thus, $\displaystyle b=a$. We see that $\displaystyle Q$ is a Sylow subgroup because it has the largest exponent of $\displaystyle p$ dividing $\displaystyle |H|$.