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Math Help - find the order of the elements in the group

  1. #1
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    find the order of the elements in the group

    find the order of the elements in the group <Z*21, .>

    i think number of element in it is 20 from {1,2,3...20}

    is it the divisors of 21 or divisors of 20?

    please give an idea.

    thank you.
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  2. #2
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    Re: find the order of the elements in the group

    The order of an element o(a) = n is the least positive integer n such that a^{n} = 1_{G}. In the case of \mathbb{Z}_{21}^{*}, we have a^{n} \equiv 1 \pmod{21}. So with 1, we clearly have o(1) = 1. With 2, we want 2^{n} \equiv 1 \pmod{21}. So ask yourself- 2^{k} * 2 \equiv 1 \pmod{21}. Solving for 2^{-1} \equiv 32 \equiv 11 \pmod{21}, and 32 = 2^{5}. So we get 2^{6} = 64 \equiv 1 \pmod{21}.

    So does that help give you some intuition in solving for the orders of the other elements?
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  3. #3
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    Re: find the order of the elements in the group

    Hi,
    Here's some discussion about your problem. I computed the orders of 8 of the elements, so you finish it up with the remaining 4.

    Thanks from macosxnerd101
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