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
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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