# 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