Group Actions and Permutation Representations

Definition 2.1.1 (G-set). Let G be a finite group. A G-set is a finite set X together with an action of G on X, namely a homomorphism a : G → Aut(X), where Aut(X) denotes the group of all bijections X → X. For convenience, we use the notation

g · x := a(g)(x).

We will use G-sets to construct interesting representations of G (see Definition 2.1.12).

Example 2.1.2 (Some G-set constructions). Let X and Y be G-sets. Then if we write g · (x, y) = (g · x, g · y), X × Y becomes a G-set. For a function f : X → Y, if we write (g · f)(x) = g · f(g−1x), then the set YX of all functions from X to Y becomes a G-set.

Definition 2.1.3 (Orbit). Let X be a G-set and x ∈ X. The G-orbit of x is the set

Gx := {g · x | g ∈ G}.

A subset O of X is said to be a G-orbit in X if it is the G-orbit of some point.

[1] Exercise 2.1.4. For two points x and y in a G-set X, write x ∼ y if y is in the G-orbit of x. Show that ‘∼’ is an equivalence relation. The equivalence classes are the G-orbits in X.

The set of G-orbits in X is usually denoted by G\X. Each G-orbit of X is a G-set in its own right.

Definition 2.1.5 (Transitive G-set). We say that a G-set X is transitive if X has only one G-orbit.

[1] Exercise 2.1.6. If X is a G-set and Y is an H-set, then X × Y is a G × H-set under the action (g, h) · (x, y) = (g · x, h · y). Describe the G × H-orbits in X × Y.