Subgroup of finite index in finitely generated group
This article describes a property that arises as the conjunction of a subgroup property: subgroup of finite index with a group property imposed on the ambient group: finitely generated group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
Definition
A subgroup of finite index in finitely generated group is a subgroup of finite index in a finitely generated group.
Facts
- Schreier's lemma shows that any subgroup of finite index in a finitely generated group is itself finitely generated, and gives an upper bound on the minimum size of generating set in terms of the corresponding value for the whole group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Subgroup of finite group |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Subgroup of finite index |