In
mathematics, a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
is called elementary amenable if it can be built up from
finite groups and
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
s by a sequence of simple operations that result in
amenable group
In mathematics, an amenable group is a locally compact topological group ''G'' carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely add ...
s when applied to amenable groups. Since finite groups and abelian groups are amenable, every elementary amenable group is amenable - however, the converse is not true.
Formally, the class of elementary amenable groups is the smallest subclass of the class of all groups that satisfies the following conditions:
*it contains all finite and all abelian groups
*if ''G'' is in the subclass and ''H'' is isomorphic to ''G'', then ''H'' is in the subclass
*it is closed under the operations of taking
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
s, forming
quotients, and forming
extensions
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
*it is closed under
directed unions.
The
Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Statement
The theorem, proven by Tits, is stated as follows.
Consequences
A linear group is not ...
implies that any amenable linear group is locally virtually solvable; hence, for linear groups, amenability and elementary amenability coincide.
References
*
{{algebra-stub
Infinite group theory
Properties of groups