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Nilpotent Group
We define the lower central series of a group
to be the filtration
of subgroups
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![$\displaystyle [G^{i-1},G],\ \ i>1,$](http://myyn.org/static/assets/mathbrowser/article/7680/images/img8.png){kind=small} |
![$ [G^{i-1},G]$](http://myyn.org/static/assets/mathbrowser/article/7680/images/img9.png){kind=small}
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
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.