The following self-contained post clarifies the definition of group action (definition 1.2, p. 3, in the book “Groups, graphs and trees: an introduction to the geometry of infinite groups” by John H. Meier) by adapting an example from chapter 7, p. 72-74, of the book “A book of abstract algebra” by Charles C. Pinter.
A rigorous definition of group action and of its group homomorphism representation was provided in a previous post. This post presents the group action on the symmetries of the square to exemplify the concept.
Consider a square with numbered vertices, as shown in the figure below. The set of vertices of the square is .
A symmetry of the square can be informally thought of as a way of moving the square so that it coincides with its former position. Every such move is fully described by its effect on the vertices, in the sense that every new vertex position coincides with a distinct former vertex position.
There are exactly 8 symmetric moves of the square, each of which can be described by a permutation of the square’s vertices. Here is the list of the 8 symmetries of the square:
- The identity
, which does not move the square.
- Clockwise rotation of the square about its centre
by an angle of
:
.
- Clockwise rotation of the square about its centre
by an angle of
:
.
- Clockwise rotation of the square about its centre
by an angle of
:
.
- Flip of the square about its diagonal
(see figure below):
.
- Flip of the square about its diagonal
:
.
- Flip of the square about its vertical axis
:
.
- Flip of the square about its horizontal axis
:
.
The set of all 8 symmetries of the square is denoted by . Define the operation
to be the function composition
for
. For example,
is the result of first flipping the square about its vertical axis
and then rotating it clockwise by
:
.
means that first flipping the square about its vertical axis
and then rotating it clockwise by
is the same as flipping the square about its diagonal
.
The set of symmetric permutations of the square along with the operation of permutation composition induces the so-called dihedral group
of the symmetries of the square.
denotes the order of group
, which is the number of elements of
. Obviously,
.
The symmetric group of
is the set of all the permutations of
, i.e. the set of all the bijective functions from
to
. Since
has
elements, the order of
is
.
Note that and that
is a subgroup of
.
Any function is a group action as long as it satisfies
for all
(identity property) and
for all
and for all
(compatibility property).
One way of picking a specific group action relies on defining the type of associated group homomorphism
in a way that respects the identity and compatibility properties of
.
The simplest possible example would be to choose the group homomorphism to be the identity function
, in which case the group action
takes the form
.
It is easy to check that the the group action , which arises by setting the group homomorphism
to be the identity function, satisfies the properties of identity and compatibility:
,
.
It is also easy to see for instance that the group action maps
to
, since
.
The group action is interpreted as the function that maps every symmetric move (permutation)
of the square and every square vertex
to the square vertex
.
The group homomorphism is the identity, so it is an injective function. As elaborated in this previous post, since
with
is injective, the group action
with
is faithful.