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

Prove that \(S_n\) is generated by the set of transpositions \(\{(1 \; 2),(2 \;3), \cdots ,(n-1 \; n ) \}\) .

Using Jacobi method and cyclic Jacobi method, find all the eigen values and corresponding eigen vectors of the following matrices:

   (a) \(\left[\begin{array}{cols} 2&1.414&4 \\1.414&6&1.414 \\ 1&1.414 &2 \end{array}\right] \)

Using Jacobi method and cyclic Jacobi method, find all the eigen values and corresponding eigen vectors of the following matrices:

   (a)  \(\left[ \begin{array}{cols} 2&3&1\\3&2&2\\ 1&2&1 \end{array}\right] \)

 In case of, (a) iterate till off diagonal elements in magnitude are less than 0.0005.

What are the advantages of Given’s method over the Jacobi’s method?

Transform the following matrices to tridiagonal system by Given’s method:

    \(\left[\begin{array}{cols} 1&2&-3 \\2&1&-1 \\-3&-1&2 \end{array}\right] \)