# 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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## Contents

## 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 is a surjective homomorphism of groups, and is a normal subgroup of . Then, is normal in .

## 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**: is a surjective homomorphism of groups, and is a normal subgroup of

**To prove**: is normal in

**Proof**: Pick and . We need to show that .

Since , there exists such that . Further, since is surjective, there exists such that . Then:

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

Now, since is normal in , , and hence , showing that .