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