Let $H$ and $E$ be normal subgroups of a group $G$ such that $$G/H \cong E.$$ Under what sort of conditions would we also have $$G/E \cong H?$$
Thanks.
Let $H$ and $E$ be normal subgroups of a group $G$ such that $$G/H \cong E.$$ Under what sort of conditions would we also have $$G/E \cong H?$$
Thanks.
This holds if $G=H\times E$, almost by definition.
Another example is $G=H \rtimes H$ for suitable $H$. $\mathbb{Z}$ works, for example, $$\langle a, b; aba^{-1}=b^{-1}\rangle$$ as does $C_6$: $$\langle a, b; a^6, b^6, aba^{-1}=b^{-1}\rangle$$ as $Aut(C_6)\cong C_2$ (because $\varphi(6)=2$). Indeed, any group $H$ such that $H\rightarrow Aut(H)$ non-trivially works.
It would be interesting to see if there exist (finite) groups $H$ and $E$ and homomorphisms $\phi$ and $\varphi$ such that $H\rtimes_{\phi} E\cong H\ltimes_{\varphi} E$. However, I haven't found any examples yet...