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.
Famous quotes containing the words elementary, amenable and/or group:
“Listen. We converse as we live—by repeating, by combining and recombining a few elements over and over again just as nature does when of elementary particles it builds a world.”
—William Gass (b. 1924)
“We must hold a man amenable to reason for the choice of his daily craft or profession. It is not an excuse any longer for his deeds that they are the custom of his trade. What business has he with an evil trade?”
—Ralph Waldo Emerson (1803–1882)
“Stripped of ethical rationalizations and philosophical pretensions, a crime is anything that a group in power chooses to prohibit.”
—Freda Adler (b. 1934)