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

1. If $a,b,c\in G$ then $\left(ab\right)c=a\left(bc\right)$.
2. There exists an identity $1\in G$,
3. 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 $$C^1\left(G\right)\supseteq C^2\left(G\right)\supseteq \dots ,\text{where}{C}^1\left(G\right)=G\text{and}{C}^{i+1}\left(G\right)=[G,C^i(G\left)\right].$$
• The derived series of $G$ is the sequence $$D^0\left(G\right)\supseteq D^1\left(G\right)\supseteq \dots ,\text{where}{D}^0\left(G\right)=G\text{and}{D}^{i+1}\left(G\right)=\left[D^i\right(G),D^i(G\left)\right].$$

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 $$\begin{array}{ccc}G\times G& ⟶& G\\ (g_1,g_2)& ⟼& g_1{g}_2\end{array}\text{and}\begin{array}{ccc}G& ⟶& G\\ g& ⟼& g^{-1}\end{array}$$ 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, Springer-Verlag, Berlin 1989. MR?????

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