# Finite verbal subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: finite subgroup and verbal subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **finite verbal subgroup** if the following equivalent conditions are satisfied:

- is a finite group and is a verbal subgroup of .
- is a finite group and is a verbal subgroup of finite type in .

## Relation with other properties

### Stronger properties

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

verbal subgroup of finite group | the whole group is finite | |FULL LIST, MORE INFO |

### Weaker properties

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

verbal subgroup of finite type | |FULL LIST, MORE INFO | |||

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

finite characteristic subgroup | finite and a characteristic subgroup: invariant under all automorphisms | |FULL LIST, MORE INFO | ||

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