Preimage of a subgroup

#group_theory

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

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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.