# Finite verbal subgroup

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

## Definition

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

1. $H$ is a finite group and is a verbal subgroup of $G$.
2. $H$ is a finite group and is a verbal subgroup of finite type in $G$.

## 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