nilpotent group

We define the lower central series of a group $G$ to be the filtration of subgroups

$$G=G^1\supset G^2\supset \mathrm{\cdots }$$

defined inductively by:

$G^1$$:=$$G,$

$G^i$$:=$$[G^{i-1},G],i>1,$

where $[G^{i-1},G]$ denotes the subgroup of $G$ generated by all commutators of the form $hk{h}^{-1}{k}^{-1}$ where $h\in G^{i-1}$ and $k\in G$. The group $G$ is said to be nilpotent if $G^i=1$ 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

$$C_0\subset C_1\subset C_2\subset \mathrm{\cdots }$$

defined by setting $C_0$ to be the trivial subgroup of $G$, and inductively taking $C_i$ to be the unique subgroup of $G$ such that $C_i/C_{i-1}$ is the center of $G/C_{i-1}$, for each $i>1$. The group $G$ is nilpotent if and only if $G=C_i$ for some $i$. Moreover, if $G$ is nilpotent, then the length of the upper central series (i.e., the smallest $i$ for which $G=C_i$) equals the length of the lower central series (i.e., the smallest $i$ for which $G^{i+1}=1$).

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 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.