# Isomorphic quotient groups

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.

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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