Isomorphic Quotient Groups: A Counterexample

[ehrenfest]

Homework Statement

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.

Homework Equations

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?

 

[StatusX]

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] so that there is actually more than one group of this order.