Isomorphic quotient groups

Suppose that $G$ is a limited team, which $H$ and also $K$ are regular subgroups of $G$ with unimportant junction, and also intend that $H$ and also $K$ are isomorphic. Is it real that the quotient teams $G/H$ and also $G/K$ are isomorphic?

32
2022-06-07 14:33:23
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Answers: 2

Let $G=\mathbb Z_2\times\mathbb Z_4$. Locate 2 subgroups in $G$ isomorphic to $\mathbb Z_2$ and also converging trivially such that the ratios of $G$ by them are not isomorphic.

35
2022-06-07 15:02:14
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As Mariano has actually revealed, the solution is a clear no .

The most effective fixing I can consider is the following: intend that $H$ and also $K$ are regular subgroups of a team $G$ such that there exists an automorphism $\varphi: G \rightarrow G$ with $\varphi(H) = K$. After that $G/H \cong G/K$: without a doubt, the isomorphism is generated by $\varphi$. Specifically, the above problem holds if $H$ and also $K$ are conjugate subgroups of $G$, which serves sufficient to be worth bearing in mind.

Keep in mind ultimately that the problem that $H \cap K = \{e\}$ appears to have absolutely nothing to do with anything: it neither aids neither injures the wanted verdict, until now as I can see.

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2022-06-07 15:02:00
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