Normality satisfies image condition

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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.