Give an example of a group $G$ with $N,M \triangleleft G$ such that $N \cong M$, but $G/M \not \cong G/N$.

Question

I'm trying to find an example of a group $G$ with $N$ and $M$ normal subgroups such that $N \cong M$, but $G/M \not \cong G/N$.

Can anybody help me with this ?

http://math.stackexchange.com/questions/151911/g-is-a-group-h-cong-k-then-is-g-h-cong-g-k?rq=1 — Metin Y., Apr 17 '13 at 20:27

score 4 · Answer 1 · answered Apr 17 '13 at 20:24

4

$N=G=\mathbb Z$, $M=2\mathbb Z$.

answered Apr 17 '13 at 20:24

Hagen von Eitzen

score 2 · Answer 2 · answered Apr 17 '13 at 20:21

2

I'm sure this is a duplicate, but $\mathbb{Z}_2\times \mathbb{Z}_4$ with subgroups generated by $(1,0)$ and $(0,2)$ respectively.

answered Apr 17 '13 at 20:21

Alexander Gruber

2 Answers2