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I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example?

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up vote 23 down vote accepted

Take $G = \mathbb{Z}_4 \times \mathbb{Z}_2$, $H$ generated by $(0,1)$, $K$ generated by $(2,0)$. Then $H \cong K \cong \mathbb{Z}_2$ but $G/H \cong \mathbb{Z}_4$ while $G/K \cong \mathbb{Z}_2 \times \mathbb{Z}_2$.

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Ah, thanks, that's a good one. Clearly my group theory is a little rusty... –  Zev Chonoles Oct 24 '10 at 20:38
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Nice example! –  anonymous Oct 25 '10 at 6:16
    
@Nate: This will also work if our $H = \mathbb{Z}_{2}$ right? –  user9413 Oct 14 '11 at 19:47
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@Chandrasekhar: I'm not sure what you're asking. You can think of $H$ as $\{0\} \times \mathbb{Z}_2 \subset \mathbb{Z}_4 \times \mathbb{Z}_2$ if you want... –  Nate Eldredge Oct 14 '11 at 21:30
    
@NateEldredge: thats precisely what i wanted. –  user9413 Oct 14 '11 at 21:53
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