# Elementary amenable group

In mathematics, a group 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 groups 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.

The Tits alternative implies that any amenable linear group is locally virtually solvable; hence, for linear groups, amenability and elementary amenability coincide.

## References

- Chou, Ching (1980). "Elementary amenable groups".
*Illinois Journal of Mathematics*.**24**(3): 396–407. MR 0573475.

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