# Abelian characteristic subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property (itself viewed as a subgroup property): Abelian group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed an **abelian characteristic subgroup** if is abelian as a group (i.e., is an abelian subgroup of and also, is a characteristic subgroup of , i.e., is invariant under all automorphisms of .

## Examples

Here are some examples of subgroups in basic/important groups satisfying the property:

Group part | Subgroup part | Quotient part | |
---|---|---|---|

A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

## Facts

- Second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group of nilpotency class , all the subgroups are abelian characteristic subgroups.
- The center of any group is an abelian characteristic subgroup.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

cyclic characteristic subgroup | cyclic group and a characteristic subgroup of the whole group | |FULL LIST, MORE INFO | ||

characteristic central subgroup | central subgroup (i.e., contained in the center) and a characteristic subgroup of the whole group | |FULL LIST, MORE INFO | ||

abelian critical subgroup | |FULL LIST, MORE INFO | |||

characteristic subgroup of center | characteristic subgroup of the center |
Characteristic central subgroup|FULL LIST, MORE INFO | ||

characteristic subgroup of abelian group | the whole group is an abelian group and the subgroup is characteristic | Characteristic central subgroup, Characteristic subgroup of center|FULL LIST, MORE INFO | ||

maximal among abelian characteristic subgroups | |FULL LIST, MORE INFO | |||

abelian fully invariant subgroup | abelian and a fully invariant subgroup -- invariant under all endomorphisms | |FULL LIST, MORE INFO |