Finite verbal subgroup

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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