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  \(o(\alpha) = LCM(n_1,n_2, \cdots ,n_r)\).