Nilpotent and solvable groups
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 27 November 2011
Nilpotent and Solvable groups
A group is a set $G$ with a multiplication such that

If $a,b,c\in G$ then
$\left(ab\right)c=a\left(bc\right)$.
 There exists an identity $1\in G$,
 Every element of $G$ is invertible.
Let $G$ be a group.
Let $x,y\in G$.
 The commutator of $x$ with $y$ is
$[x,y]=xy{x}^{1}{y}^{1}$.

The lower central series of $G$ is the sequence
$${C}^{1}\left(G\right)\supseteq {C}^{2}\left(G\right)\supseteq \dots \phantom{\rule{0.5em}{0ex}},\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}{C}^{1}\left(G\right)=G\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}{C}^{i+1}\left(G\right)=[G,{C}^{i}(G\left)\right].$$
 The derived series of $G$ is the sequence
$${D}^{0}\left(G\right)\supseteq {D}^{1}\left(G\right)\supseteq \dots \phantom{\rule{0.5em}{0ex}},\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}{D}^{0}\left(G\right)=G\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}{D}^{i+1}\left(G\right)=\left[{D}^{i}\right(G),{D}^{i}(G\left)\right].$$
Let $G$ be a group.
 The group $G$ is abelian
if $[G,G]=\left\{1\right\}$.
 The group $G$ is nilpotent if
there exists $n\in {\mathbb{Z}}_{>0}$
such that ${C}^{n}\left(G\right)=\left\{1\right\}$.
 The group $G$ is solvable if
there exists $n\in {\mathbb{Z}}_{>0}$
such that ${D}^{n}\left(G\right)=\left\{1\right\}$.

The radical $R\left(G\right)$ of a
Lie??? group $G$ is the largest connected solvable normal subgroup of $G$.
WHY IS THE LIE IN THIS DEFNITION??
Nondiscrete groups
 A topological group is a topological space $G$
which is also a group such that multiplication and inversion
$$\begin{array}{ccc}G\times G& \u27f6& G\\ ({g}_{1},{g}_{2})& \u27fc& {g}_{1}{g}_{2}\end{array}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\begin{array}{ccc}G& \u27f6& G\\ g& \u27fc& {g}^{1}\end{array}$$ are morphisms of topological spaces, i.e. continuous maps.

A Lie group is a smooth manifold with a group structure such that multiplication and inversion are morphisms of smooth manifolds, i.e. smooth maps.
 A complex Lie group is a complex analytic manifold with a group structure such that multiplication and inversion are morphisms of complex analytic manifolds, i.e. holomorphic maps.
 A linear algebraic group is an affine algebraic variety which is also a group such that multiplication and inversion are morphisms of affine algebraic varieties.
 A group scheme is a scheme which is also a group such that multiplication and inversion are morphisms of schemes.
Notes and References
These basic definitions need to be combined with another page somewhere. The reference below needs fixing.
References
[BouTop]
N. Bourbaki,
General Topology, Chapter VI, SpringerVerlag, Berlin 1989.
MR?????
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