mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 iden ...

is called elementary amenable if it can be built up from

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

s 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 commut ...

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 additi ...

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 subgroup ...

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 * Ex ...

*it is closed under

directed union In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...

s. 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 am ...

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