# Amenable discrete group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A discrete group $G$ is termed an amenable discrete group if there exists a function that assigns to each subset of $G$ a real number in $[0,1]$ such that:

• The number associated with the whole of $G$ is $1$
• The function is finitely additive on subsets. In other words, the number associated with a disjoint union of finitely many subsets is the sum of the numbers associated with each of the subsets
• The function is left-invariant, viz., the value associated with a subset $S$ is the same as the value associated with $gS$

Often, when dealing with an abstract group, we use the term amenable group for amenable discrete group (in other words, we assume that the discrete topology is put over the group).

## Metaproperties

### Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A finite direct product of amenable discrete groups is again an amenable discrete group.

### Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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A subgroup of an amenable group is amenable. Further information: Amenability is subgroup-closed

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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A quotient of an amenable group is amenable. Further information: Amenability is quotient-closed