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!

Homework Help: 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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook