# Subgroup of finite index in finitely generated group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: subgroup of finite index with a group property imposed on theambient 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

## Contents

## 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 |