If $H$ and $K$ are normal subgroups of $G$ and $G/H \cong K$, does this imply that $G/K \cong H$?
I wasn't able to find a counterexample or to prove that the implication is true. I would appreciate any help with this question. Thank you!
2026-03-26 17:33:34.1774546414
If $H$ and $K$ are normal subgroups of $G$ and $G/H \cong K$, does this imply that $G/K \cong H$?
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Counterexample: take $G=Q=\{ 1, -1, i, -i, j, -j, k, -k\}$ the quaternion group of order $8$. Take $H=\langle i \rangle$ and $K=Z(Q)=\{1, -1\}$. $G/K$ in non-cyclic, where $H$ is.