A group is locally finite when every subgroup that is a finitely generated group is finite. It is a torsion group, since the cyclic subgroup generated by one element must be finite. A restricted direct sum of groups with finite factors is locally finite. A locally finite group can be infinite, while a locally finite finitely generated group is finite by definition.
For with locally finite, take a finitely generated subgroup . Its image in is finite, so has finite index in and is finitely generated by Schreier's lemma. Local finiteness of makes that intersection finite. Thus itself is finite, proving local finiteness of .
// Target: algebra.bigb
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A **locally finite group** is a type of group in the field of abstract algebra. Specifically, a group \( G \) is called locally finite if every finite subset of \( G \) generates a finite subgroup of \( G \). In other words, for any finite subset \( S \) of \( G \), the subgroup generated by \( S \), denoted by \( \langle S \rangle \), is finite.