# Elementary amenable 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 group is said to be **elementary amenable** if it can be built from finite groups and Abelian groups by the following operations:

- Taking subgroups
- Taking quotient groups
- Taking group extensions
- Taking directed unions

Since each of these operations preserves amenability, every elementary amenable group is an amenable discrete group (viz, its amenable when viewed with the discrete topology).