# Retraction-invariant characteristic subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: retraction-invariant subgroup and characteristic subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **retraction-invariant characteristic subgroup** if it satisfies the following two conditions:

- It is a retraction-invariant subgroup, i.e., any retraction of the whole group sends the subgroup to itself.
- It is a characteristic subgroup, i.e., any automorphism of the whole group sends the subgroup to itself.

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Fully invariant subgroup | invariant under all endomorphisms | |FULL LIST, MORE INFO |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Characteristic subgroup | invariant under all automorphisms | |FULL LIST, MORE INFO | ||

Normal subgroup | invariant under all inner automorphisms | Retraction-invariant normal subgroup|FULL LIST, MORE INFO | ||

Retraction-invariant normal subgroup | normal and retraction-invariant | |FULL LIST, MORE INFO |