### 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)=C*^{G}(x) := {gxg^{-1} | g ÎG },

and
*G*_{x}=C_{G}(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
*G*_{U}=N_{G}(U) := { g | gU=Ug } .

last changed: January 19, 2005