Group theory

Definition (group)

A (multiplicative) group is a set equipped with a binary operation denoted satisfying the following properties:

• (associativity)
• (existence of identity)
• (existence of inverse) The symbol for the operation is often omitted. For example, one would commonly write instead of .

”Mattress” group

The mattress group is the group of all possible movements of a mattress in a bed frame. The task at hand is the following: imagine that you have a mattress, and you want to be sure that it does not wear down. To do this, you need to move it around in the bed frame. The mattress is rectangular and fits the bed frame perfectly.

Then you can move the mattress in the following ways:

1. Rotate the mattress degrees horizontally (flipping the head and legs, but keeping the same part facing up): let us call this rotation .
2. Rotate the mattress degrees vertically (flipping the mattress over, but keeping the head and leg position in the same place): let us call this rotation .

The identity element in this group is “do nothing"" (i.e., keep the mattress in the same position).

The group operation is simply the composition of the two rotations. For example, if you first rotate the mattress degrees horizontally and then rotate it degrees vertically, you will have a new position of the mattress. The group is closed under this operation, meaning that any combination of these two operations will yield another position of the mattress in the bed frame.

Performing twice, we will have the mattress in the same position as before. So . The same holds for : .

Have we exhausted all possible movements of the mattress? No, we could have first performed and then .

Question

What is the result of performing and then ? What is the result of performing and then ? Do you get the same answer?

Answer

In this case . Try to imagine (or even draw!) this. The resulting group has elements: with the following multiplication table:

*	e	h	v	hv
e	e	h	v	hv
h	h	e	hv	v
v	v	hv	e	h
hv	hv	v	h	e

Symmetries of a regular polygon

A regular polygon is a polygon with all sides of equal length and all angles of equal measure. The symmetries of a regular polygon are the transformations that preserve its shape and size (one would wish to be more careful here, but we keep this definition at an intuitive level instead of formalizing it). These transformations include, for example, rotations and reflections.

The symmetries of a regular polygon form a group under the operation of composition. The identity element is the transformation that leaves the polygon unchanged. The inverse of a transformation is the transformation that undoes it. The notation for this group is ,

Exercise

List all the elements in and . Write down the multiplication table for these groups.

A symmetry of a regular polygon can be interpreted as a permutation of the vertices of the polygon. For example, if we have a regular hexagon with vertices labeled (counterclockwise), then a rotation of degrees counterclockwise can be represented as the permutation

This means that the vertex moves to the position of vertex , vertex moves to the position of vertex , and so on. The inverse permutation is the one that undoes this transformation, which in this case is a rotation of degrees counterclockwise (or degrees clockwise). This can be represented as the permutation

The composition of two symmetries can be represented as the product of their corresponding permutations. For example, if we have two rotations and , we can represent their composition as the product of their corresponding permutations. The resulting permutation will be the one that corresponds to the symmetry obtained by performing both transformations in sequence. For example, if we have as the rotation of degrees counterclockwise and as the rotation of degrees counterclockwise, we can represent their composition as

Keep in mind that we use left-to-right agreement for permutation multiplication. This means that the permutation on the left is applied first, and then the one on the right. Both left-to-right and right-to-left exist in the literature and are used, and you can use either in the final if you used it in your preparation: both will be accepted.