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Efficient way to find which element of Z*n has largest order?

  1. Feb 13, 2013 #1
    Hey, I had two separate questions. (When I say Z*(n) I'm denoting the multiplicative group of integers mod n, namely, the units of Z(n))

    First off, I know that if n is prime, that Z*(n) is cyclic. But this is not a biconditional statement. Is there any theorem which tells me conditions under which Z*(n) is not cyclic? I can't just say n is not prime so Z*(n) is not cyclic. Right?

    Second question - if I know that Z*(n) is cyclic, then I know it has a generator. But other than going through each and every element and multiplying them out, is there an efficient way to find out which elements are generators? Also, in the case that Z*(n) isn't cyclic (like say Z*(12)) is there a way to find which element has the highest order?

    If anyone can point me to the proper theorems, I would be extremely grateful!
     
  2. jcsd
  3. Feb 13, 2013 #2

    AlephZero

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    Start by thinking about the case where n = pq where p and q are both prime. (For example, n = 6)

    When you see what is going on, generailze it to the case where n has more than two prime factors.
     
  4. Feb 13, 2013 #3

    mathwonk

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    Aleph Zero makes a good suggestion. It is not so trivial in general however, when there are repeated factors. There is a discussion starting on p. 48. of these notes, of when the group is cyclic:

    http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]

    In fact your question as to precisely how to identify generators is perhaps even less trivial. There may even be some open questions surrounding that matter, as I recall.
     
    Last edited by a moderator: May 6, 2017
  5. Feb 13, 2013 #4
    Thank you guys for the links and the responses!

    I also found this:

    "Zn*, the multiplicative group modulo n, is cyclic if and only if n is 1 or 2 or 4 or pk or 2pk for an odd prime number p and k ≥ 1."

    But I did not see a proof provided for it. I think this might tie in with what you were saying AlephZero
     
  6. Feb 13, 2013 #5

    mathwonk

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    if you look at the links i provided you will find complete proofs of these facts.
     
  7. Feb 13, 2013 #6
    Yeah I have spent the last couple hours looking at the doc you provided and some other documents I've hunted down. Very helpful! I need to purchase a good number theory text book to have on hand it seems! Thank you again!
     
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