Locally finite group

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

Definition

Symbol-free definition

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

Every subgroup of it that is finitely generated, is in fact finite.
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 $g_{1}, g_{2}, \dots, g_{n} \in G$ 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:

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

where the inclusion maps are multiplication by $p$ maps. Equivalently, it can be thought of as the multiplicative group of the union of all $(p^{n})^{t h}$ roots of unity in the complex numbers for all $n$ .

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.

Formalisms

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In terms of the locally operator

This property is obtained by applying the locally operator to the property: finite group
View other properties obtained by applying the locally operator

Relation with other properties

Stronger properties

Finite group

Weaker properties

Periodic group

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

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of a locally finite group is locally finite.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a locally finite group is locally finite.

Direct products

This group property is restricted direct product-closed, viz., a restricted direct product of groups, each having the property, also has the property.
View more such properties

A restricted direct product of locally finite groups is locally finite. In particular, a direct product of finitely many locally finite groups is locally finite.

For full proof, refer: Local finiteness is restricted direct product-closed