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prove that the group U(n^2 -1) is not cyclic

  1. Feb 29, 2012 #1
    Sorry if I formatted this thread incorrectly as its my first post ^^

    1. The problem statement, all variables and given/known data

    For every integer n greater than 2, prove that the group U(n^2 - 1) is not cyclic.


    2. Relevant equations



    3. The attempt at a solution

    I've done a problem proving that U(2^n) is not cyclic when n >3, but I'm failing to make a parallel.

    How does one find the order of (n^2 -1)? Is this information even needed to solve this problem?
     
  2. jcsd
  3. Feb 29, 2012 #2

    Deveno

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    you might want to prove some easier things first:

    suppose 8|k, that is:

    k = 2tm, where t ≥ 3.

    then φ(k) = φ(2t)φ(m), so

    U(k) ≅ U(2t) x U(m).

    use this to show U(k) cannot be cyclic, as it has a non-cyclic subgroup.

    now, consider even n and odd n for n2-1 separately.
     
  4. Feb 29, 2012 #3
    I have learned that property of Euler's totient function, but not the one about U(ab) = U(a) x U(b). I've come across it online, but have yet to use it in class.

    Is there another method?
     
  5. Feb 29, 2012 #4

    Deveno

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    U(ab) isn't always isomorphic to U(a) x U(b).

    for example U(4) is cyclic of order 2, but U(2) x U(2) has but a single element: (1,1).

    it IS true, if a and b are co-prime.

    this is actually a consequence of the chinese remainder theorem:

    if gcd(m,n) = 1, then [a]mn→([a]m,[a]n) is a group isomorphism of (Zmn,+) with (Zm,+) x (Zn,+).

    it's easy to check that [a]mn→([a]m,[a]n) is a ring homomorphism as well, hence Zmn and Zm x Zn have isomorphic groups of units (when...gcd(m,n) = 1. this is key).

    i find it odd, that you wouldn't use this result, since it builds on the result of your previous problem.
     
  6. Feb 29, 2012 #5
    It may be because it's still fairly early in the course. I learned the Chinese Remainder Theorem in a number theory class, but have yet to encounter it in my current one. We are going to study rings after the upcoming exam.

    The previous problem was in a different chapter of the text I believe. This question is supposed to be a "review."
     
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