Assignment:
a) Let a be the m-cycle (123.....m). how to show that a^i is also an m-cycle if and only if i is relatively prime to m. Here a is an element of group G that generates the m-cycle.
b) How to prove that the order of an element in Sn equals the least common multiple of the lengths of the cycles in its cycle decomposition.
Provide complete and step by step solution for the question and show calculations and use formulas.