1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Non simplicity of a group

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data
    Let G be a group with |G|= mp where p is prime and 1<m<p. Prove that G is not simple

    2. Relevant equations

    3. The attempt at a solution
    I have proven the existence of a subgroup H that has order p(via Cauchy's Theorem), but I don't know how to use the representation of G on cosets of H or another method to somehow deduce that H is a normal subgroup of G thus forcing G to be not simple.
  2. jcsd
  3. Oct 16, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Let G act on the cosets of H by left translation. This induces a homomorphism f:G->S_[G:H]=S_m (why?). Since kerf sits in H (why?), we can consider [H:kerf]. Since |H|=p, it follows that either [H:kerf]=1 or [H:kerf]=p (why?). If it's the former, we're done (why?). So suppose that [H:kerf]=p and deduce a contradiction.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - simplicity group Date
Please check my (simple) proof. Skeptical of its simplicity Sep 5, 2011
Proving Non-Simplicity May 13, 2009