In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 is called elementary amenable if it can be built up from finite groups and abelian groups by a sequence of simple operations that result in amenable group

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...

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 subgroups, forming quotients, and forming extensions
• it is closed under directed unions.