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)=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 } .

Examples