Using Cayley's theorem to find the order of a power in a group

Abstract

It is shown how Cayley's theorem can be used to prove the formula for the order of a power of an element of finite order in a group. Reasoning with disjoint cycles leads to a proof that depends on elementary number theory in some new ways, leading naturally to some new connections involving least common multiples, greatest common divisors and minima.

Keywords:

• Cayley's theorem
• group
• order
• disjoint cycles
• greatest common divisor
• least common multiple
• prime-power factorization
• congruence modulo n