# Retract is transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., retract) satisfying a subgroup metaproperty (i.e., transitive subgroup property)

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 retract |Get facts that use property satisfaction of retract | Get facts that use property satisfaction of retract|Get more facts about transitive subgroup property

## Contents

## Statement

Suppose are groups such that is a retract of and is a retract of , then is a retract of .

## Definitions used

### Retract

`Further information: Retract`

A subgroup of a group is termed a retract of if there exists a homomorphism such that for all .

## Proof

**Given**: A group , a retract of , a retract of .

**To prove**: is a retract of .

**Proof**: Let be a retraction, i.e., is a homomorphism such that for all . Let be a retraction, i.e., is a homomorphism such that for all .

Now consider the composite map . We want to argue that this is a retraction. Consider . Then, by construction , so . Since , , and so we get . Thus, is a retraction, and thus is a retract of .