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.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	local finiteness is subgroup-closed	If $G$ is a locally finite group, and $H$ is a subgroup of $G$ , then $H$ is also locally finite.
quotient-closed group property	Yes	local finiteness is quotient-closed	If $G$ is a locally finite group, and $H$ is a normal subgroup of $G$ , then the quotient group $G / H$ is also locally finite.
extension-closed group property	Yes	local finiteness is extension-closed	If $G$ is a group and $H$ is a normal subgroup such that both $H$ and $G / H$ are locally finite, then $G$ is also locally finite.
restricted direct product-closed group property	Yes	local finiteness is restricted direct product-closed	If $G_{i}, i \in I$ are all locally finite groups, so is the restricted external direct product of the $G_{i}$ s.

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