Two factor groups are isomorphic

Question

Let $G$ be a finite Abelian group and let $N_1, N_2$ be two isomorphic subgroups of $G$. To show that the factor groups $G/N_1$ and $G/N_2$ are isomorphic.

What I tried so far is the following. Let $g$ be the isomorphism from $N_1$ onto $N_2$. We define $f:G \to G/N_2$ by $f(x) = g(x)g(N_1)$ for all $x $ in $G$.

However I am unable to show $f$ is onto group homomorphism. May be my approach is incorrect. Please guide me.

Thanks in well advance.

Answer 1

What you want to prove is false. Let $G = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_4$. Consider $N_1= \mathbb{Z}_2 \times \mathbb{Z}_2 \times \{0\}$ and $N_2 = \{0\} \times \mathbb{Z}_2 \times \langle 2 \rangle$. Then $N_1 \cong N_2 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ yet $G/N_1 \cong \mathbb{Z_4}$ and $G/N_2 \cong \mathbb{Z_2} \times \mathbb{Z_2}$.