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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== *Ching Chou (1980), ''Elementary amenable groups'', Illinois J. Math. 24, p. 396-407. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「elementary amenable group」の詳細全文を読む スポンサード リンク
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