Examples of Groups - The Symmetry Group of A Square - Dihedral Group of Order 8

Dihedral Group of Order 8

Groups are very important to describe the symmetry of objects, be they geometrical (like a tetrahedron) or algebraic (like a set of equations). As an example, we consider a glass square of a certain thickness (with a letter "F" written on it, just to make the different positions discriminable). In order to describe its symmetry, we form the set of all those rigid movements of the square that don't make a visible difference (except the "F"). For instance, if you turn it by 90° clockwise, then it still looks the same, so this movement is one element of our set, let's call it a. We could also flip it horizontally so that its underside become up. Again, after performing this movement, the glass square looks the same, so this is also an element of our set and we call it b. Then there's of course the movement that does nothing; it's denoted by e.

Now if you have two such movements x and y, you can define the composition x ∘ y as above: you first perform the movement y and then the movement x. The result will leave the slab looking like before.

The point is that the set of all those movements, with composition as operation, forms a group. This group is the most concise description of the square's symmetry. Chemists use symmetry groups of this type to describe the symmetry of crystals.

Let's investigate our squares symmetry group some more. Right now, we have the elements a, b and e, but we can easily form more: for instance a ∘ a, also written as a2, is a 180° degree turn. a3 is a 270° clockwise rotation (or a 90° counter-clockwise rotation). We also see that b2 = e and also a4 = e. Here's an interesting one: what does a ∘ b do? First flip horizontally, then rotate. Try to visualize that a ∘ b = b ∘ a3. Also, a2 ∘ b is a vertical flip and is equal to b ∘ a2.

This group of order 8 has the following Cayley table:

	e	b	a	a2	a3	ab	a2b	a3b
e	e	b	a	a2	a3	ab	a2b	a3b
b	b	e	a3b	a2b	ab	a3	a2	a
a	a	ab	a2	a3	e	a2b	a3b	b
a2	a2	a2b	a3	e	a	a3b	b	ab
a3	a3	a3b	e	a	a2	b	ab	a2b
ab	ab	a	b	a3b	a2b	e	a3	a2
a2b	a2b	a2	ab	b	a3b	a	e	a3
a3b	a3b	a3	a2b	ab	b	a2	a	e

For any two elements in the group, the table records what their composition is.

Here we wrote "a3b" as a short hand for a3 ∘ b.

Mathematicians know this group as the dihedral group of order 8, and call it either Dih₄, D₄ or D₈ depending on what notation they use for dihedral groups. This was an example of a non-abelian group: the operation ∘ here is not commutative, which you can see from the table; the table is not symmetrical about the main diagonal.

The dihedral group of order 8 is isomorphic to the permutation group generated by (1234) and (13).