Explore: Nilpotent Groups

nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups

groups of polynomial growth, as those groups which have nilpotent subgroups of finite index. The growth rate of a group is a well-defined notion from asymptotic

the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras)

finite p-groups are solvable, as all finite p-groups are nilpotent. In particular, the quaternion group is a solvable group given by the group extension

an element x {\displaystyle x} of a ring R {\displaystyle R} is called nilpotent if there exists some positive integer n {\displaystyle n} , called the

understanding finite solvable groups as extensions of nilpotent groups by faithful automorphism groups of nilpotent groups. In a nilpotent group, every chief factor

such a group is 1-dimensional. Like solvable groups, nilpotent groups are too messy to classify except in a few small dimensions. Simple Lie groups are sometimes

1962) for nilpotent groups and later extended by Bertram Kostant, Louis Auslander, Lajos Pukánszky and others to the case of solvable groups. Roger Howe

undecidability of the elementary theory of finite groups, of free nilpotent groups, of free soluble groups and many others. He also proved that the class

theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups.[citation needed]