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 in...