   Examples

### Examples

The first bunch of examples which illustrate these concepts will show that various important group theoretical structures can be considered as orbits or stabilizers:
Examples: If G denotes a group, then
• G acts on itself by left multiplication : G ´G -> G :(g,x) -> g ·x. This action is called the (left) regular representation of G, it is obviously transitive, and all the stabilizers are equal to the identity subgroup  {1 }.
• G acts on itself by conjugation : G ´G -> G :(g,x) -> g ·x ·g-1. The orbits of this action are the conjugacy classes of elements, and the stabilizers are the centralizers of elements:
G(x)=CG(x) := {gxg-1 | g ÎG },
and
Gx=CG(x) := {g | gxg-1=x }.
• If U denotes a subgroup of G (in short: U £G), then G acts on the set G/U := {xU | x ÎG } of its left cosets as follows:
G ´G/U -> G/U :(g,xU) -> gxU.
This action is transitive, and the stabilizer of xU is the subgroup xUx-1 which is conjugate to U.
• G acts on the set L(G) := {U | U £G } of all its subgroups by conjugation : G ´L(G) -> L(G) :(g,U) -> g U g-1. The orbits of this action are the conjugacy classes of subgroups , and the stabilizers are the normalizers:
G(U) = [~U] := { gUg-1 | g ÎG },
and
GU=NG(U) := { g | gU=Ug } .

harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001   Examples