Verbal subgroup of finitely generated type
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
A subgroup of a group is termed a verbal subgroup of finitely generated type if it satisfies the following equivalent conditions:
- It is the subgroup generated by the image of the word map for a single word.
- It is the subgroup generated by the union of the images of the word maps for finitely many words.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| verbal subgroup of finite type | |FULL LIST, MORE INFO | |||
| finitely generated verbal subgroup | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| purely definably generated subgroup | subgroup generated by a purely definable subset | |FULL LIST, MORE INFO | ||
| characteristic subgroup | invariant under all automorphisms | |FULL LIST, MORE INFO |