Orders of Elements in a Group

cos 2π n + i sin 2π n is an example of an element of C× with order n. If G is a finite group, every g ∈ G has finite order. The proof is as follows. Since the set of powers {ga : a ∈ Z} is a subset of G and the exponents a run over all integers, an infinite set, there must be a repetition: ga = gb for some a < b in Z. Then gb−a = e, so g has finite order. (Taking the contrapositive, if g has infinite order its integral powers have no repetitions: ga = gb =⇒ a = b.) There are three questions about elements of finite order that we want to address: (1) When g ∈ G has a known finite order, how can we tell when two powers gk and g` are the same directly in terms of the exponents k and `? (2) When g has finite order, how is the order of a power gk related to the order of g? (3) When two elements g1 and g2 of a group have finite order, how is the order of their product g1g2 related to the orders of g1 and g2?