Let $G$ be a group, and $M$, $N$ be normal subgroups such that $M \leq N$. If $G/N \cong G/M$ then need it be true that $N = M$?

Question

Let $G$ be a group, and $M$, $N$ be normal subgroups such that $M \leq N$. Suppose that $G/N \cong G/M$, then need it be true that $N = M$? My gut instinct is yes, but unsure how to go about showing this.

I am aware that without the stipulation that $M \leq N$, the statement does not hold. I believe I have seen this statement used (without justification) in a paper and I'm just having trouble justifying it myself.

Thanks in advance.

Answer 1

No, it is not true. Take $G=\mathbb Z\times\mathbb Z\times\cdots$ (a countable product), $N=\mathbb Z\times\{0\}\times\{0\}\times\cdots$ and $M$ the trivial subgroup.