mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

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 commu ...

s by a sequence of simple operations that result in

amenable group Amenable may refer to: * Amenable group * Amenable species * Amenable number * Amenable set See also * Agreeableness Agreeableness is the trait theory, personality trait of being kind, Sympathy, sympathetic, cooperative, warm, honest, strai ...

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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

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 (proof theory) * Extension (predicate logic), the set of tuples of values t ...

*it is closed under directed unions. The

Tits alternative In mathematics, the Tits alternative, named after 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

* Infinite group theory Properties of groups {{group-theory-stub