# Are quotient groups unique up to isomorphism

By this post, it seems quotient groups are unique up to isomorphism. is it correct? More clearly

Let $G$ be a group and let $K,N\unlhd G$ be isomorphic normal subgroups. Are $\frac{G}{N}$ and $\frac{G}{K}$ isomorphic?

• Not necessarily. There is a counterexample with $G$ abelian of order $8$. – Derek Holt Nov 28 '14 at 3:33
• @Derek So the last part of that answer is not correct!? – user795571 Nov 28 '14 at 3:40
• @user795571 Notice in the answer that $\phi(g)$ is undefined if $g\in G-N$. – Joe Johnson 126 Nov 28 '14 at 3:41
• Every subgroup $n\Bbb Z\lhd \Bbb Z$ is isomorphic to $\Bbb Z$. – Pedro Tamaroff Nov 28 '14 at 3:45
• For a finite example, $\mathbb{Z}^4 \oplus \mathbb{Z}^2/e \oplus \mathbb{Z}^2 \neq \mathbb{Z}^4 \oplus \mathbb{Z}^2/ \mathbb{Z}^2 \oplus e$ (Following Derek Holt's hint) – Tim kinsella Nov 28 '14 at 4:00

$\mathbb{Z}/2\mathbb{Z}\neq\mathbb{Z}/3\mathbb{Z}$