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Math Help - finite groups of complex numbers

  1. #1
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    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.
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  2. #2
    Senior Member Shanks's Avatar
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    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.
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