(a) Let be a finite group, and let . Define the order of and show it is finite. Show that if is conjugate to , then and have the same order.
(b) Show that every can be written as a product of disjoint cycles. For , describe the order of in terms of the cycle decomposition of .
(c) Define the alternating group . What is the condition on the cycle decomposition of that characterises when ?
(d) Show that, for every has a subgroup isomorphic to .