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
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1$\begingroup$ See this one $\endgroup$ – Mikasa Oct 29 '13 at 10:27
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No, this is not true. Take $G = \Bbb Z$, with $H_1 = 2\Bbb Z$ and $H_2 = 3\Bbb Z$ as subgroups.
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$\begingroup$ @PtF I do not know of any classification of when this holds. In general, if $G$ is infinite, the quotient groups need not even have the same order (as demonstrated here). If $G$ is finite, then the quotient groups need to have the same order, so at least if $|G|/|H_i|$ is prime, you know it holds. $\endgroup$ – Arthur Oct 29 '13 at 10:34
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1$\begingroup$ @PtF It holds if there is an automorphism of $G$ that carries $H_1$ to $H_2$. $\endgroup$ – Zhen Lin Oct 29 '13 at 10:53
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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 $G/\langle b\rangle\cong\mathbb{Z}_4$.
answered Oct 29 '13 at 10:29
