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Show G abelian

  1. Sep 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Let G be a finite group and let I ={g in G: g^2 = e} \ {e} be its subset of involutions. Show that G is abelian if card(I) => (3/4)card(G).

    2. Relevant equations



    3. The attempt at a solution


    I don't really know how to proceed with this problem and to make use of 3/4. I know that the set I is precisely the elements of order 2 in the group G and all elements within I U {e} commute; but I just don't know how to show the whole group commutes. Any help on how to proceed with the problem is highly appreciated. Thanks.
     
  2. jcsd
  3. Sep 14, 2009 #2

    Dick

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    I'll give you a hint while I'm trying to figure this one out myself. You don't know that I commutes, do you? It's perfectly possible that a^2=e and b^2=e and ab is not equal to ba, isn't it? If you HAVE figured out how to prove {e} U I is commutative, then you would be done. That makes it a subgroup. Use Lagrange's theorem.
     
  4. Sep 15, 2009 #3

    Dick

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    For future reference, I think I finally got this one. Consider an element a in I. Now look at the |G| equations aG=G. Show if ab=c and b and c are also in I, then b commutes with a. Count how many equations in aG=G are of that form and draw a conclusion about the size of the centralizer subgroup of a. The rest is an easy exercise. That was bothering me.
     
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