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== *{{cite journal|first1=Ching|last1= Chou|date=1980|title=Elementary amenable groups|journal=Illinois Journal of Mathematics|volume=24|issue=3|pages=396–407|doi= 10.1215/ijm/1256047608|url=https://www.projecteuclid.org/euclid.ijm/1256047608|mr=573475|s2cid= 122441593|doi-access=free}}
Category:Infinite group theory Category:Properties of groups
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