Exercises

1. Let $G$ be a finite group and let $p\vert\, \vert G\vert$. Suppose $P$ is a $p$-Sylow subgroup of $G$. Prove that any conjugate of $P$, $gPg^{-1}$, is also a $p$-Sylow subgroup of $G$.

2. Let $G$ be a finite group and $N \lhd G$ such that $\vert N\vert$ is a power of a prime $p$. Prove that $N$ is contained in every $p$-Sylow subgroup of $G$. (HINT: Use Theorems 4.3.6 and Proposition 8.3.6.)

3. Let $G$ be a finite group and $P$ be a $p$-Sylow subgroup of $G$. Prove that if $x \in N_G(P)$ and $o(x)$ is a power of $p$, then $x \in P$. (HINT: Same as for exercise 2.)

4. Let $G$ be a finite group and $P$ be a $p$-Sylow subgroup of G. Prove that $P$ is the only $p$-Sylow subgroup of $G$ contained in $N_G(P)$. (HINT: Use exercise 3.)