Preimage of a subgroup
Suppose
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Suppose $\phi:G\to G'$ is a surjective homomorphism, $H\leq G$ is a subgroup containing $\ker(\phi)$, and $H'=\phi(H)$. Prove $\phi^{-1}(H')=H$, where $\phi^{-1}(H')=\{g\in G\,\mid\, \phi(g)\in H'\}$. Make sure to state explicitly where each hypothesis is used.