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

    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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