1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Conjugacy class with two elements in G implies that G is not simple

  1. Jan 7, 2010 #1
    1. The problem statement, all variables and given/known data
    If some conjugacy class of an element in a group G contains precisely two elements, show that G cannot be a simple group.

    3. The attempt at a solution
    This question was longer, with two questions before this one which I could answer and which probably lead to the answer on this question.

    I showed that for an element x in G the elements of G which commute with x, form a subgroup C(x) of G, called the centralizer. I also proved that the size of the conjugacy class of x is equal to the number of left cosets of C(x) in G. But now for the last question, I need a hint.
  2. jcsd
  3. Jan 7, 2010 #2
    What can you say about a subgroup of index 2?
  4. Jan 7, 2010 #3
    A subgroup H<G of index two is normal, since if you take x in G, H and xH partition G and H and Hx partition G, which gives Hx=xH, and H is normal. If I have a normal subgroup of index 2, that means that the order of this subgroup is half of the order of G, which cannot be {e} or G, unless G={e}. This is not the case, since a conjugacy class of G contains two distinct elements. Therefore H is a normal subgroup which is not {e} or all of G, hence G is not simple.

    Is this correct? Thanks for helping me out in your free time!
  5. Jan 7, 2010 #4
    But how do you know you have a subgroup of index two? Otherwise you are correct.
  6. Jan 7, 2010 #5
    Because I proved earlier that the index of the centralizer C(x) is the same as the order of the conjugacy class of x, and C(x) is a subgroup.
    Thanks thanks :)
  7. Jan 7, 2010 #6
    Exactly, I figured you knew why, you just hadn't explicitly stated it. Don't thank me too much; I just gave a nudge in the right direction.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook