# Sylow p-subgroups

• Nov 9th 2008, 06:45 PM
dori1123
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.
• Nov 9th 2008, 07:33 PM
ThePerfectHacker
Quote:

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 $\displaystyle |H|$.