# Abelian verbal subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: verbal 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 verbal subgroup** if is an abelian group in its own right (i.e., it is an abelian subgroup of ) and is a verbal subgroup of .

## Examples

- In an abelian group, the verbal subgroups are precisely the power subgroups, see verbal subgroup of abelian group.
- In a nilpotent group, second half of lower central series of nilpotent group comprises abelian groups, so all of these are abelian verbal subgroups.
- In a solvable group, the penultimate term of the derived series is an abelian verbal subgroup.

### Examples in small finite groups

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:

## Relation with other properties

### Stronger properties

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

verbal subgroup of abelian group | |FULL LIST, MORE INFO |

### Weaker properties

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

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

abelian characteristic subgroup | abelian and a characteristic subgroup: invariant under all automorphisms | Abelian fully invariant subgroup|FULL LIST, MORE INFO | ||

abelian normal subgroup | abelian and a normal subgroup: invariant under all inner automorphisms | Abelian fully invariant subgroup|FULL LIST, MORE INFO |