nilpotent group

We define the lower central seriesPlanetmathPlanetmath of a group G to be the filtrationPlanetmathPlanetmath of subgroupsMathworldPlanetmathPlanetmath


defined inductively by:

G1 := G,
Gi := [Gi-1,G],i>1,

where [Gi-1,G] denotes the subgroup of G generated by all commutatorsPlanetmathPlanetmath of the form hkh-1k-1 where hGi-1 and kG. The group G is said to be nilpotent if Gi=1 for some i.

Nilpotent groups can also be equivalently defined by means of upper central series. For a group G, the upper central series of G is the filtration of subgroups


defined by setting C0 to be the trivial subgroup of G, and inductively taking Ci to be the unique subgroup of G such that Ci/Ci-1 is the center of G/Ci-1, for each i>1. The group G is nilpotent if and only if G=Ci for some i. Moreover, if G is nilpotent, then the length of the upper central series (i.e., the smallest i for which G=Ci) equals the length of the lower central series (i.e., the smallest i for which Gi+1=1).

The nilpotency class or nilpotent class of a nilpotent group is the length of the lower central series (equivalently, the length of the upper central series).

Nilpotent groups are related to nilpotent Lie algebras in that a Lie group is nilpotent as a group if and only if its corresponding Lie algebraMathworldPlanetmath is nilpotent. The analogyMathworldPlanetmath extends to solvable groupsMathworldPlanetmath as well: every nilpotent group is solvable, because the upper central series is a filtration with abelianMathworldPlanetmath quotients.

Title nilpotent group
Canonical name NilpotentGroup
Date of creation 2013-03-22 12:47:50
Last modified on 2013-03-22 12:47:50
Owner djao (24)
Last modified by djao (24)
Numerical id 8
Author djao (24)
Entry type Definition
Classification msc 20F18
Defines nilpotent
Defines upper central series
Defines lower central series
Defines nilpotency class
Defines nilpotent class