I am trying to understand a specific step in Artin's outline of a proof of the "correspondence theorem." The setup is as follows.
- $\varphi: G \to \mathcal{G}$ is a surjective group homomorphism with kernel $K$.
- He claims that there is a bijection between subgroups containing $K$ and subgroups of $\mathcal{G}$.
- $H$ is a subgroup of $G$ which contains $K$ and $\mathcal{H}$ is a subgroup of $\mathcal{G}$.
I proved that $\varphi(H)$ is a subgroup of $\mathcal{G}$, that $\varphi^{-1} (\mathcal{H})$ is a subgroup of $G$ containing $K$, and then $\mathcal{H}$ is a normal subgroup of $\mathcal G$ if and only if $\varphi^{-1} (\mathcal{H})$ is a normal subgroup of $G$. MY problem is the next bullet point. Artin labels it "bijectivity of the correspondence," but I don't quite understand how it proves bijectivity:
(bijectivity of the correspondence) $\varphi(\varphi^{-1} (\mathcal{H})) = \mathcal{H}$ and $\varphi^{-1} (\varphi(H)) = H$.
I don't have a problem with proving these as a matter of set equality, though I don't understand how this gives bijectivity. Certainly a function is bijective if and only if it admits an inverse, but $\varphi^{-1}$ is only the inverse image here. So I'm not sure why this gets around having to prove injectivity and surjectivity from the definitions.