NilpotentGroup 2002-06-16 23:54:18 2006-11-01 11:40:07 Definition nilpotent upper central series lower central series nilpotency class nilpotent class % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here We define the {\em lower central series} of a group $G$ to be the filtration of subgroups $$ G = G^1 \supset G^2 \supset \cdots $$ defined inductively by: \begin{eqnarray*} G^1 & := & G, \\ G^i & := & [G^{i-1},G],\ \ i>1, \end{eqnarray*} 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 {\em 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 {\em 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 \emph{nilpotency class} or \emph{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.