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Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements

1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$

2) If $G/N_{1}\cong G/N_{2}$ then $N_{1}\cong N_{2}$.

Thanks for the help.

Shaun
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TheNumber23
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    Or try harder to solve it yourself. There is an abelian group of order 8 which provides a counterexample to both 1 and 2. – Derek Holt Aug 14 '13 at 21:09
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    @YACP: It’s not a particularly easy thing to search for; I doubt that I’d have managed to find a relevant question without your strong suggestion that one was there to be found to keep me trying different searches. You really ought to have supplied a pointer. – Brian M. Scott Aug 14 '13 at 23:49
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    Take a look at this question. – Brian M. Scott Aug 14 '13 at 23:50

1 Answers1

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Hint: Take $G=\mathbb{Z}_4\times \mathbb{Z}_2$. This group provides a basis for counter-examples to both of your statements.

user1729
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