Normality satisfies image condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., image condition)
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Statement

Property-theoretic statement

The subgroup property of being normal satisfies the image condition: the image of a normal subgroup under any surjective homomorphism is also normal.

Statement with symbols

Suppose $\varphi :G\to H$ is a surjective homomorphism of groups, and $N$ is a normal subgroup of $G$. Then, $\varphi (N)$ is normal in $H$.

Generalizations

This result is part of a more general result called the fourth isomorphism theorem (also called the lattice isomorphism theorem or correspondence theorem).

Proof

Given: $\varphi :G\to H$ is a surjective homomorphism of groups, and $N$ is a normal subgroup of $G$

To prove: $\varphi (N)$ is normal in $H$

Proof: Pick $a\in \varphi (N)$ and $b\in H$. We need to show that $bab^{-1}\in \varphi (N)$.

Since $a\in \varphi (N)$, there exists $g\in N$ such that $\varphi (g)=a$. Further, since $\varphi$ is surjective, there exists $h\in G$ such that $\varphi (h)=b$. Then:

$bab^{-1}=\varphi (h)\varphi (g)\varphi (g)^{-1}=\varphi (hgh^{-1})$

(where the second step uses the fact that $\varphi$ is a homomorphism).

Now, since $N$ is normal in $G$, $hgh^{-1}\in N$, and hence $\varphi (hgh^{-1})\in \varphi (N)$, showing that $bab^{-1}\in N$.