# nilpotent group

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

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

defined inductively by:

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

where $[G^{i-1},G]$ denotes the subgroup of $G$ generated by all commutators of the form $hkh^{-1}k^{-1}$ 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

 $C_{0}\subset C_{1}\subset C_{2}\subset\cdots$

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.

Title nilpotent group NilpotentGroup 2013-03-22 12:47:50 2013-03-22 12:47:50 djao (24) djao (24) 8 djao (24) Definition msc 20F18 nilpotent upper central series lower central series nilpotency class nilpotent class