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

## 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 $G$.

## 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 $G$.

• Subgroup of Finite Index Contains a Normal Subgroup of Finite Index Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.   Proof. The group $G$ acts on the set of left cosets $G/H$ by left multiplication. Hence […]
• Normal Subgroups, Isomorphic Quotients, But Not Isomorphic Let $G$ be a group. Suppose that $H_1, H_2, N_1, N_2$ are all normal subgroup of $G$, $H_1 \lhd N_2$, and $H_2 \lhd N_2$. Suppose also that $N_1/H_1$ is isomorphic to $N_2/H_2$. Then prove or disprove that $N_1$ is isomorphic to $N_2$.   Proof. We give a […]