Subgroup of finite index in finitely generated group

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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