# Amenable discrete group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A discrete group is termed an **amenable discrete group** if there exists a function that assigns to each subset of a real number in such that:

- The number associated with the whole of is
- 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 is the same as the value associated with

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).

## Relation with other properties

### Stronger properties

- Finite group, equipped with the counting measure. This is the measure associating to every subset the ratio of its cardinality to the cardinality of the group
- Solvable group:
`For full proof, refer: Solvable implies amenable`

## 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

View a complete list of subgroup-closed group properties

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

View a complete list of quotient-closed group properties

A quotient of an amenable group is amenable. `Further information: Amenability is quotient-closed`