If $G$ is a group with $|G|=p^n$ , where $p$ is a prime, and if $0 \le k \le n$ , show that $G$ contains a normal subgroup of order $p^k$  .

Let $H$ be a proper subgroup of a finite p-group $G$ . If $H=p^s$ , then prove that there is a subgroup of $G$ order $p^{s+1}$ containing $H$ .

Prove that a finite p-group cannot be simple unless it has order p

Let G be a finite group and p a prime. Show that a normal p-subgroup of G is contained in every Sylow p-subgroup of G.

If $P$ is a normal Sylow p-subgroup of $G$ and $H \le G$, prove that $P \cap H$ is a unique Sylow p-subgroup of $H$.

Let $H$ be a subgroup of a finite group $G$ and $p$ a prime. Prove that $|Syl_p(H)\ \le |Syl_p(G)|$ .

If $G$ is a finite group and $p \in Syl_p(G)$, show that $N_G(N_G(P))=N_G(P)$ .

Let $G$ be a finite group and $H \le G$. Suppose that $P \in Syl_p(H)$ . If $N_G(P) \subseteq H$, show that $P \in Syl_p(G)$ .

Let $G$ be a finite group and $P \in Syl_p(G)$ , and let $H$ be a subgroup of $G$ containing $N_G(P)$ . Prove that $N_G(H) =H$ .

Show that, if $\sigma \in S_n$ is a cycle of length $r$, then $o(\sigma) =r$ .

Let $\sigma \in S_n$ be a product of disjoint cycles $\alpha_1,\alpha_2, \cdots ,\alpha_r$ of lengths $n_1,n_2, \cdots ,n_r,$ respectively. Prove that  $o(\alpha) = LCM(n_1,n_2, \cdots ,n_r)$.