# Thread: finite groups of complex numbers

1. ## finite groups of complex numbers

Hi, I'm having trouble proving the following.
Any finite group of complex numbers with respect to multiplication is a group of all the roots of unity of degree n for some n
My approach to this problem to show that any complex number other than a root of unity cannot be in a finite group; however, I'm not sure how to proceed.

2. Let G (not {0}) be a finite group of complex numbers with respect to multiplucation, |G| = n .
we have : if x belons to G , |x| = 1. for ,otherswise, the collection of all the powers of x is a infinite subset of G , Contradiction!
Since G is finite , Let g be the element such that Arg(g) (positive) is the smallest among G. then G=<g>, g is of oder n. Thus G is a group of all the roots of unity of degree n.