Amenable discrete group: Difference between revisions
No edit summary |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 3: | Line 3: | ||
==Definition== | ==Definition== | ||
A [[group]] <math>G</math> is termed an '''amenable discrete group''' if there exists a function that assigns to each subset of <math>G</math> a real number in <math>[0,1]</math> such that: | A [[discrete group]] <math>G</math> is termed an '''amenable discrete group''' if there exists a function that assigns to each subset of <math>G</math> a real number in <math>[0,1]</math> such that: | ||
* The number associated with the whole of <math>G</math> is <math>1</math> | * The number associated with the whole of <math>G</math> is <math>1</math> | ||
| Line 9: | Line 9: | ||
* The function is left-invariant, viz., the value associated with a subset <math>S</math> is the same as the value associated with <math>gS</math> | * The function is left-invariant, viz., the value associated with a subset <math>S</math> is the same as the value associated with <math>gS</math> | ||
Often, when dealing 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== | ==Relation with other properties== | ||
| Line 23: | Line 23: | ||
A finite direct product of amenable discrete groups is again an amenable discrete group. | A finite direct product of amenable discrete groups is again an amenable discrete group. | ||
{{S-closed}} | |||
A subgroup of an amenable group is amenable. {{further|[[Amenability is subgroup-closed]]}} | |||
{{Q-closed}} | |||
A quotient of an amenable group is amenable. {{further|[[Amenability is quotient-closed]]}} | |||
Latest revision as of 23:20, 28 September 2010
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 properties
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 is termed an amenable discrete group if there exists a function that assigns to each subset of a real number in such that:
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
- 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