5Group actions

IA Groups

5.1 Group acting on sets

Definition

(Group action)

.

Let

X

be a set and

G

be a group. An action of

G

on X is a homomorphism ϕ : G → Sym X.

This means that the homomorphism

ϕ

turns each element

g ∈ G

into a

permutation of X, in a way that respects the group structure.

Instead of writing ϕ(g)(x), we usually directly write g(x) or gx.

Alternatively, we can define the group action as follows:

Proposition.

Let

X

be a set and

G

be a group. Then

ϕ

:

G → Sym X

is a

homomorphism (i.e. an action) iff

θ

:

G × X → X

defined by

θ

(

g, x

) =

ϕ

(

g

)(

x

)

satisfies

0. (∀g ∈ G)(x ∈ X) θ(g, x) ∈ X.

1. (∀x ∈ X) θ(e, x) = x.

2. (∀g, h ∈ G)(∀x ∈ X) θ(g, θ(h, x)) = θ(gh, x).

This criteria is almost the definition of a homomorphism. However, here we

do not explicitly require

θ

(

g, ·

) to be a bijection, but require

θ

(

e, ·

) to be the

identity function. This automatically ensures that

θ

(

g, ·

) is a bijection, since

when composed with

θ

(

g

−1

, ·

), it gives

θ

(

e, ·

), which is the identity. So

θ

(

g, ·

)

has an inverse. This is usually an easier thing to show.

Example.

(i)

Trivial action: for any group

G

acting on any set

X

, we can have

ϕ

(

g

) = 1

X

for all g, i.e. G does nothing.

(ii) S

n

acts on {1, · · · n} by permutation.

(iii) D

2n

acts on the vertices of a regular n-gon (or the set {1, · · · , n}).

(iv)

The rotations of a cube act on the faces/vertices/diagonals/axes of the

cube.

Note that different groups can act on the same sets, and the same group can

act on different sets.

Definition

(Kernel of action)

.

The kernel of an action

G

on

X

is the kernel of

ϕ, i.e. all g such that ϕ(g) = 1

X

.

Note that by the isomorphism theorem,

ker ϕ C G

and

G/K

is isomorphic to

a subgroup of Sym X.

Example.

(i) D

2n

acting on {1, 2 · · · n} gives ϕ : D

2n

→ S

n

with kernel {e}.

(ii)

Let

G

be the rotations of a cube and let it act on the three axes

x, y, z

through the faces. We have

ϕ

:

G → S

3

. Then any rotation by 180

◦

doesn’t

change the axes, i.e. act as the identity. So the kernel of the action has

at least 4 elements:

e

and the three 180

◦

rotations. In fact, we’ll see later

that these 4 are exactly the kernel.

Definition (Faithful action). An action is faithful if the kernel is just {e}.