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
