Locally finite group

Symbol-free definition
A group is said to be locally finite if it satisfies the following equivalent conditions:


 * 1) Every subgroup of it that is finitely generated, is in fact finite.
 * 2) It is the direct limit of a directed system of finite groups.

Definition with symbols
A group $$G$$ is said to be locally finite if for any finite subset $$g_1, g_2, \ldots, g_n \in G$$ the group generated by the $$g_i$$s is a finite group.

Examples

 * For a prime number $$p$$, the $$p$$-quasicyclic group is a locally finite group. It is obtained as a direct limit of inclusions:

$$0 \to \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \dots \to \mathbb{Z}/p^n\mathbb{Z} \to $$

where the inclusion maps are multiplication by $$p$$ maps. Equivalently, it can be thought of as the multiplicative group of the union of all $$(p^n)^{th}$$ roots of unity in the complex numbers for all $$n$$.
 * The group of rational numbers modulo integers $$\mathbb{Q}/\mathbb{Z}$$ is an example of a locally finite group.
 * The finitary symmetric group on a possibly infinite set is locally finite, because any finite subset of the group has finite support and hence lives inside the symmetric group on a finite subset.

Opposite properties

 * Finitely generated group: A group that is both finitely generated and locally finite must be finite.

Dual properties

 * Residually finite group
 * Profinite group: A topological, completed version of residually finite