# Transitive normality satisfies image condition

From Groupprops

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

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about transitively normal subgroup |Get facts that use property satisfaction of transitively normal subgroup | Get facts that use property satisfaction of transitively normal subgroup|Get more facts about image condition

## Contents

## Statement

### Statement with symbols

Suppose is a transitively normal subgroup of a group . Suppose is a surjective homomorphism of groups. Then, is a transitively normal subgroup of .

## Related facts

### Similar facts about similar properties

- Central factor satisfies image condition
- SCAB satisfies image condition
- Direct factor satisfies image condition
- Centrality satisfies image condition

### Related facts about transitively normal subgroups

## Facts used

## Proof

**Given**: A group , a subgroup . A surjective homomorphism of groups. . is a normal subgroup of .

**To prove**: is normal in .

**Proof**:

- is normal in and : Let be the restriction of . Then, is a surjective homomorphism by definition, and fact (1) yields that is normal in . Further, clearly surjects to , since is surjective. But by definition, so is normal in and .
- is normal in : From the previous step, is normal in . By assumption, is transitively normal in . Thus, must be normal in .
- is normal in </math>K</math>: This follows from fact (2).