The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal

Normal Subgroups Problems and Solutions in Group Theory

Problem 116

Let $G$ and $G’$ be groups and let $f:G \to G’$ be a group homomorphism.
If $H’$ is a normal subgroup of the group $G’$, then show that $H=f^{-1}(H’)$ is a normal subgroup of the group $G$.

 
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Proof.

We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup in $G$ is left as an exercise.)
For any element $g \in G$ and $h \in H$, we have
\[f(ghg^{-1})=f(g)f(h)f(g)^{-1}\] since $f$ is a group homomorphism.

Since $f(g) \in G’$, $f(h)\in H’$, and $H’$ is normal in $G’$, we see that
\[f(ghg^{-1})=f(g)f(h)f(g)^{-1} \in H’.\] Thus by the definition of $H$, the element $ghg^{-1} \in H$.
This proves that $H$ is a normal subgroup in $G$.


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