Math 2.0

Abstract harmonic analysis Definition

Elementary amenable group

Builds on GroupAbelian groupSubgroupQuotient groupGroup extensionDirect limitAmenable group

Definition

The class EGEG of elementary amenable groups is the smallest class of groups with these properties.

  1. EGEG contains every finite group and every abelian group.
  2. If G∈EGG \in EG and HH is a subgroup of GG, then H∈EGH \in EG.
  3. If G∈EGG \in EG and NN is a normal subgroup of GG, then the quotient G/NG/N is in EGEG.
  4. If NN is a normal subgroup of GG with N∈EGN \in EG and G/N∈EGG/N \in EG, then G∈EGG \in EG; that is, EGEG is closed under extensions.
  5. If GG is the union of a family of subgroups in EGEG that is directed by inclusion, then G∈EGG \in EG. So EGEG is closed under directed unions.

A group is elementary amenable if it belongs to EGEG.

Remarks

The intersection of all classes with properties 1 to 5 again has these properties, so the smallest such class exists. Every elementary amenable group is amenable, since the class of amenable groups has properties 1 to 5.

Example: the infinite dihedral group D∞=ℤ⋊ℤ/2D_\infty = \Z \rtimes \Z/2 has the normal subgroup ℤ\Z with quotient ℤ/2\Z/2. Both are abelian, so D∞D_\infty is elementary amenable by property 4. In the same way, every solvable group is elementary amenable, by induction on the length of its derived series.

Non-example: the free group F2F_2 on two generators is not elementary amenable, since it is not amenable. The Grigorchuk group is amenable but not elementary amenable, because it has intermediate growth, and no elementary amenable finitely generated group has intermediate growth.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Abelian group, concept, no evidence levelAbelian groupconceptAmenable group, concept, no evidence levelAmenable groupconceptDirect limit, concept, no evidence levelDirect limitconceptGroup, concept, no evidence levelGroupconceptGroup extension, concept, no evidence levelGroup extensionconceptQuotient group, concept, no evidence levelQuotient groupconceptSubgroup, concept, no evidence levelSubgroupconceptElementary amenable group, definition, no evidence levelElementary amenable g…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Abelian group concept
  2. Amenable group concept
  3. Direct limit concept
  4. Group concept
  5. Group extension concept
  6. Quotient group concept
  7. Subgroup concept
  8. Elementary amenable group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Elementary_amenable_group
About this page
TagEB7FFD48W8
URIm2://EB7FFD48W8
Kinddefinition
WritingW0 stub
MSC43A07, 20F65, 20F19
DefinesElementary amenable group
UsesGroup, Abelian group, Subgroup, Quotient group, Group extension, Direct limit, Amenable group
Changed3 November 2026

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