1. Not finding help here? Sign up for a free 30min 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!

Proving Non-Simplicity

  1. May 13, 2009 #1
    1. The problem statement, all variables and given/known data
    Let G be a finite group. Prove that if some conjugacy class has exactly two elements then G can't be simple


    2. Relevant equations



    3. The attempt at a solution
    I originally proved this accidentally assuming that G is abelian, which it isn't. So say x and y are conjugate then we have conjugacy classes {e} and {x,y}. I know that the size of a conjugacy class has to divide the order of a group so |G| is even and must be greater than 2. I'm having trouble showing the existence of a normal subgroup though.
     
  2. jcsd
  3. May 13, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If G has a conjugacy class with two elements then it has a subgroup of order |G|/2. Look at the centralizer of each element of the conjugacy class. If G has a subgroup H of order |G|/2, H must be normal. Look at left and right cosets.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook