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Subgroup of finite index in finitely generated group

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