IDA: Interactive Document on Algebra

Let G be a group. Three fundamental examples of permutation representations of G into Sym(X) with X = G are:

The left regular representation

L_g : G -> G , x -> g x.
Thus, the map L_g is left multiplication by g on G.
The right regular representation

R_g : G -> G , x -> xg^-1.
Thus, the map R_g is right multiplication by g^-1 on G.
The conjugation representation

C_g : G -> G , x -> gxg^-1.
Thus, the map C_g is conjugation by g on G.

The next proposition establishes that they are indeed permutation representations.

Proposition

The maps L, R, C are permutation representations of G on G

Next we study the kernels of these representations. We need the following subgroup of G. The center of G is the subgroup

Z(G) = {g

G | xg = gx for all x

of G.

Theorem

The kernels of L and R are trivial. In particular, every group is isomorphic with a permutation group.

The kernel of C is the center of G.

The permutation representations L, R and C all have degree |G|. In the next pages, methods for constructing lower-degree permutation representations will appear.