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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


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    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.
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