Chapter 19

Show that the centraliser of \((12)(34)(56)\) in \(S(6)\) is isomorphic to the wreath product \(C_{2}\) pwr \(S(3)\).

Let \(G\) be a group with precisely two distinct prime divisors, \(p\) and \(q\). Given that the Sylow \(p\)-subgroup is a normal subgroup of \(G\), show that \(G\) is a semidirect product.

Let \(G\) be a semidirect product of a group \(N\) with two elements by a subgroup \(H\). Show that \(G\) is an internal direct product of \(N\) and \(H\).