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}.