Dih4 as 2D point group, D4, [4], (*4•), order 4, with a 4-fold rotation and a mirror generator.
Dih4 in 3D dihedral group D4, [4,2]+, (422), order 4, with a vertical 4-fold rotation generator order 4, and 2-fold horizontal generator
Cycle graph of Dih4 a is the clockwise rotation and b the horizontal reflection.Cayley graph of Dih4A different Cayley graph of Dih4, generated by the horizontal reflection b and a diagonal reflection c
As an example, we consider a glass square of a certain thickness with a letter "F" written on it to make the different positions distinguishable. In order to describe its symmetry, we form the set of all those rigid movements of the square that do not make a visible difference (except the "F"). For instance, if an object turned 90° clockwise still looks the same, the movement is one element of the set, for instance a. We could also flip it around a vertical axis so that its bottom surface becomes its top surface, while the left edge becomes the right edge. 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. The movement that does nothing is denoted by e.
Given two such movements x and y, it is possible to define the composition x ∘ y as above: first the movement y is performed, followed by the movement x. The result will leave the slab looking like before.
The set of all those movements, with composition as the operation, forms a group. This group is the most concise description of the square's symmetry.
Generating the group
Applying two symmetry transformations in succession yields a symmetry transformation. 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.
A horizontal flip followed by a rotation, a ∘ b is the same as b ∘ a3. Also, a2 ∘ b is a vertical flip and is equal to b ∘ a2.
The two elements a and bgenerate the group, because all of the group's elements can be written as products of powers of a and b.
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 shorthand for a3 ∘ b.
In mathematics this group is known as the dihedral group of order 8, and is either denoted Dih4, D4 or D8, depending on the convention.
This is an example of a non-abelian group: the operation ∘ here is not commutative, which can be seen from the table; the table is not symmetrical about the main diagonal.
There are five different groups of order 8. Three of them are abelian: the cyclic groupC8 and the direct products of cyclic groups C4×C2 and C2×C2×C2. The other two, the dihedral group of order 8 and the quaternion group, are not.[3]
The dihedral group of order 8 is isomorphic to the permutation group generated by (1234) and (13). The numbers in this table come from numbering the 4! = 24 permutations of S4, which Dih4 is a subgroup of, from 0 (shown as a black circle) to 23.
Normal subgroup
This version of the Cayley table shows that this group has one normal subgroup shown with a red background. In this table r means rotations, and f means flips. Because the subgroup is normal, the left coset is the same as the right coset.
The elements e, r1, r2, and r3 form a subgroup, highlighted in red (upper left region). A left and right coset of this subgroup is highlighted in green (in the last row) and yellow (last column), respectively.
;_subgroup_of_S4.svg)