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 ◁ 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}.