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I think it is an easy question, but for some reasons I am confused. Thank you for your help! Let $G$, $H_1$ and $H_2$ be finite groups. If

$G/H_1\cong G/H_2$, is it true that $H_1\cong H_2$?

If not, do you know a counterexample?

Thank you!

1 Answers 1

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No. Take $G=\mathbf{Z}_2 \oplus\mathbf{Z}_4$. Let $H_1 = \mathbf{Z}_2\times\langle 2\rangle$, and $H_2=\{0\}\times\mathbf{Z}_4$. Then $G/H_1 \cong \mathbf{Z}_2\cong G/H_2$, but $H_1$ is the Klein $4$-group and $H_2$ is cyclic of order $4$.

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    Ok thank you, I meant$G$cyclic. Thank you for your answer!2012-05-17