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.