Proving inverse image is a normal subgroup

Let $a\in\Phi^{-1}(H')$ and $g\in G$. Note that $\Phi(a)\in H'$ and since $H'$ is normal $$\Phi (gag^{-1})=\Phi(g) \Phi(a) \Phi(g)^{-1}\in H'.$$ Hence $gag^{-1}\in\Phi^{-1}(H')$.

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Group Theory

Normal Subgroups

Proof Explanation

Group Homomorphism

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