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Math Help - Elements of order

  1. #1
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    Red face Elements of order

    Hi guys, if i say 2 is a primitive root in mod 19, how can i find all the elements of order 6 in Z*19, without finding the order of each element? Thank you for the help
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  2. #2
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    Re: Elements of order

    Hi Audrey,
    If you know a little about groups, it's not too hard. But some computation is necessary, I think. See the attachment:

    Elements of order-mhfgroups16.png
    Thanks from audrey
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  3. #3
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    Re: Elements of order

    thank you john, i get to know where to start!
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  4. #4
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    Re: Elements of order

    In general, if we have a primitive root in (\mathbb{Z}_p)^{\times}, we know this group is cyclic, and cyclic groups have some special properties.

    In particular, we know that if G = \langle a \rangle, the order of a^k is \frac{|G|}{\text{gcd}(k,|G|)}.

    In this example, we have |G| = 18, so we seek k such that 18/(gcd(k,18)) = 6, so that gcd(k,18) = 3. How many such k are there?

    Well, clearly k = 3 works, and of course, we need only check the following: 6,9,12,and 15. We can eliminate the even numbers straight-away, leaving just 9 and 15. Since 9|18, that leaves just 15, and indeed gcd(15,18) = 3.

    So we have two such k: 3 and 15. So the elements of order 6 are precisely: 23 = 8, and 215 = 85 = (82)(82)(8) = (7)(7)(8) = (7)(18) = 7(-1) = -7 = 12.
    Last edited by Deveno; September 4th 2013 at 10:00 AM.
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