If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
Let $G$ be a group and $H_1$ and $H_2$ be two subgroups of $G$. Is it true that if $H_1$ is isomorphic to $H_2$ then $G/H_1$ is isomorphic to $G/H_2$? If that is not true then in what conditions that holds? Any help will be useful.. Thanks
No, this is not true. Take $G = \Bbb Z$, with $H_1 = 2\Bbb Z$ and $H_2 = 3\Bbb Z$ as subgroups.
It is not true. A counter-example: Let $G=\mathbb{Z}_4\oplus\mathbb{Z}_2=\langle a\rangle \oplus\langle b\rangle$. Then $G/\langle a^2\rangle\cong\mathbb{Z}_2\oplus\mathbb{Z}_2$ but $G/\langle b\rangle\cong\mathbb{Z}_4$.