defined inductively by:
where denotes the subgroup of generated by all commutators of the form where and . The group is said to be nilpotent if for some .
Nilpotent groups can also be equivalently defined by means of upper central series. For a group , the upper central series of is the filtration of subgroups
defined by setting to be the trivial subgroup of , and inductively taking to be the unique subgroup of such that is the center of , for each . The group is nilpotent if and only if for some . Moreover, if is nilpotent, then the length of the upper central series (i.e., the smallest for which ) equals the length of the lower central series (i.e., the smallest for which ).
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 algebra is nilpotent. The analogy extends to solvable groups as well: every nilpotent group is solvable, because the upper central series is a filtration with abelian quotients.
|Date of creation||2013-03-22 12:47:50|
|Last modified on||2013-03-22 12:47:50|
|Last modified by||djao (24)|
|Defines||upper central series|
|Defines||lower central series|