## 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      $\left[x,y\right]=xy{x}^{-1}{y}^{-1}$.
• The lower central series of $G$ is the sequence $C1(G) ⊇ C2(G) ⊇… , where C1(G)=G and Ci+1(G) =[G, Ci(G)] .$
• The derived series of $G$ is the sequence $D0(G) ⊇ D1(G) ⊇… , where D0(G)=G and Di+1(G) =[ Di(G), Di(G)] .$

Let $G$ be a group.

• The group $G$ is abelian if $\left[G,G\right]=\left\{1\right\}$.
• The group $G$ is nilpotent if there exists $n\in {ℤ}_{>0}$ such that ${C}^{n}\left(G\right)=\left\{1\right\}$.
• The group $G$ is solvable if there exists $n\in {ℤ}_{>0}$ 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 $G×G ⟶ G (g1,g2) ⟼ g1g2 and G ⟶ G g ⟼ g-1$ 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?????