# Endomorphism kernel is quotient-transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., endomorphism kernel) satisfying a subgroup metaproperty (i.e., quotient-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 endomorphism kernel |Get facts that use property satisfaction of endomorphism kernel | Get facts that use property satisfaction of endomorphism kernel|Get more facts about quotient-transitive subgroup property

## Contents

## Statement

Suppose are groups such that is an endomorphism kernel in and the quotient group is an endomorphism kernel in . (Note that it makes sense to take quotients because endomorphism kernel implies normal). Then, is an endomorphism kernel in .

## Related facts

### Similar facts about similar properties

- Normality is quotient-transitive
- Characteristicity is quotient-transitive
- Complemented normal is quotient-transitive

### Opposite facts about endomorphism kernel

## Definitions used

We use the following definition of endomorphism kernel: a normal subgroup of a group is an endomorphism kernel if there exists a subgroup of isomorphic to the quotient group .

## Facts used

- Third isomorphism theorem: This basically tells us that .

## Proof

**Given**: Groups such that is an endomorphism kernel in and is an endomorphism kernel in .

**To prove**: is an endomorphism kernel in .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Let be a subgroup of isomorphic to and let be an isomorphism between them. | is an endomorphism kernel in | Definition-direct | ||

2 | Let be a subgroup of isomorphic to . | is an endomorphism kernel in . | Definition-direct | ||

3 | is isomorphic to . | Fact (1) | , normal in , normal in (normality follows from being endomorphism kernels) | Fact-direct | |

4 | is a subgroup of isomorphic to . | Steps (2), (3) | step-combination direct | ||

5 | is a subgroup of (and hence of ) isomorphic to | Steps (1), (2) | is an isomorphism, hence it maps subgroups to subgroups isomorphic to them. Apply this to . | ||

6 | is a subgroup of isomorphic to . | Steps (4), (5) | step-combination direct. | ||

7 | is an endomorphism kernel in . | Step (6) | definition-direct. |