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Groups whose order have order two

  1. Oct 17, 2009 #1
    I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative.


    Also, Consider Zn = {0,1,....,n-1}
    a. show that an element k is a generator of Zn if and only if k and n are relatively prime.

    b. Is every subgroup of Zn cyclic? If so, give a proof. If not, provide an example.
     
  2. jcsd
  3. Oct 17, 2009 #2
    The forum rules state that you must show your attempt at a solution.
     
  4. Oct 17, 2009 #3
    I would show an attempt if I knew how to start it.
     
  5. Oct 17, 2009 #4
    Here are some hints:
    a) If k generates Zn, then k must have order n. But the order of <k> = n / gcd(k,n).
    b) Are subgroups of cyclic groups necessarily cyclic?
     
  6. Oct 18, 2009 #5
    b) as far as i know, subgroups of cyclic groups are always cyclic. but to be honest, I do not know if we are allowed to assume Zn is cyclic to begin with. It states that Zn = {0,1,...(n-1)}.

    a) so because k generates Zn, generating k (x) amount of times will give us all the elements in Zn?
     
  7. Oct 18, 2009 #6

    Hurkyl

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    I'm going to assume you've already spent a good amount of time experimenting with algebraic expressions to which you can apply x²=1 in creative ways.

    If you haven't, you really should have.


    So if the full question is too hard for you, then try a simpler problem first.

    First, try to prove it in the case where G has zero generators.
    Now, try to prove it in the case where G has one generators.
    Now, try to prove it in the case where G has two generators.
    Figure it out yet? No? Then try three generators....
     
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