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Math Help - Number of Generators

  1. #1
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    Question Number of Generators

    The number of generators of a group G is the number of positive integers < n which are relatively prime to n. How can this be proved?
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  2. #2
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    Quote Originally Posted by MathBird View Post
    The number of generators of a group G is the number of positive integers < n which are relatively prime to n. How can this be proved?
    The above is true only when G is finite and has an element of order = order of the group = n. i.e is G is cyclic.

    G then = {e, a, a^2, a^3 ,......., a^n-1}

    An element is a generator IFF its order = n

    Let a^k be generator. If order of a^k = m then m is smallest number such that n | mk

    n | mk implies (n/gcd(n,k)) | m

    So smallest such m = n/gcd(n,k). For this to be n gcd(n,k) should be 1. Hence k is relatively prime to n (and obviously it is < n)
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  3. #3
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    Thank you very much.
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