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Isomorphic quotient groups

  1. Feb 4, 2008 #1
    1. The problem statement, all variables and given/known data
    Let H and K be normal subgroups of a group G. Give an example showing that we may have H isomorphic to K while G/H is not isomorphic to G/K.

    2. Relevant equations

    3. The attempt at a solution
    I don't want to look in the back of my book just yet. Can someone give me a hint?
  2. jcsd
  3. Feb 4, 2008 #2


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    Homework Helper

    If G and K are conjugates, then its easy to show that G/H and G/K are isomorphic, so you need to look for non-conjugate subgroups (more generally, this is true if there is an automorphism of G taking H to K). You also obviously need to pick [G:H] so that there is actually more than one group of this order.
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