Dihedral group of order 8

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

In mathematics, D₄ (sometimes alternatively denoted by D₈) is the dihedral group of degree 4 and order 8. It is the symmetry group of a square. ^{[1] [2]}

Symmetries of a square

As an example, consider a square of a certain thickness with the letter "F" written on it to make the different positions distinguishable. In order to describe its symmetry, one can 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, ⁠${\displaystyle a}$⁠. One 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 square looks the same, so this is also an element of our set, which is called it ⁠${\displaystyle b}$⁠. The movement that does nothing is denoted by ⁠${\displaystyle e}$⁠.

The square's initial position
(the identity transformation)

Rotation by 90° anticlockwise

Rotation by 180°

Rotation by 270°

Diagonal NW–SE reflection

Horizontal reflection

Diagonal NE–SW reflection

Vertical reflection

Generating the group

With composition as the operation, the set of all those movements forms a group. This group is the most concise description of the square's symmetry. Applying two symmetry transformations in succession yields a symmetry transformation. For instance a${\displaystyle \circ }$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${\displaystyle \circ }$b is the same as b${\displaystyle \circ }$a³. Also, a²${\displaystyle \circ }$b is a vertical flip and is equal to b${\displaystyle \circ }$a².

The two elements a and b generate the group, because all of the group's elements can be written as products of powers of a and b.

Cycle graph of Dih₄
a is the clockwise rotation
and b the horizontal reflection.

Cayley graph of Dih₄

A different Cayley graph of Dih₄, generated by the horizontal reflection b and a diagonal reflection c

This group of order 8 has the following Cayley table:

${\displaystyle \circ }$	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${\displaystyle \circ }$b. This group has 5 conjugacy classes, they are ${\displaystyle \{e\},\{a^{2}\},\{b,ba^{2}\},\{ba,ba^{3}\},{\text{ and }}\{a,a^{3}\}}$.

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 ${\displaystyle \circ }$ 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 group C₈ 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]

Permutation representation

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.

The action of a rotation or diagonal reflection on the corners of a square, numbered consecutively, can be obtained by the two permutations (1234) and (13), respectively. As the positions of all four corners uniquely determine the element of the symmetries of the square used to obtain those positions, and so the group of symmetries of a square is isomorphic to the permutation group generated by (1234) and (13).

Matrix representation

The symmetries of an axis-aligned square centered at the origin can be represented by signed permutation matrices, acting on the plane by multiplication on column vectors of coordinates ${\displaystyle {\bigl [}{\begin{smallmatrix}x\\y\end{smallmatrix}}{\bigr ]}}$. The identity transformation is represented by the identity matrix ${\displaystyle {\bigl [}{\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}{\bigr ]}}$. Reflections across a horizontal and vertical axis are represented by the two matrices ${\displaystyle {\bigl [}{\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}}{\bigr ]}}$ and ${\displaystyle {\bigl [}{\begin{smallmatrix}-1&0\\0&1\end{smallmatrix}}{\bigr ]}}$, respectively, and the two diagonal reflections are represented by the matrices ${\displaystyle {\bigl [}{\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}{\bigr ]}}$ and ${\displaystyle {\bigl [}{\begin{smallmatrix}0&-1\\-1&0\end{smallmatrix}}{\bigr ]}}$. Rotations clockwise by 90°, 180°, and 270° are represented by the matrices ${\displaystyle {\bigl [}{\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}{\bigr ]}}$, ${\displaystyle {\bigl [}{\begin{smallmatrix}-1&0\\0&-1\end{smallmatrix}}{\bigr ]}}$, and ${\displaystyle {\bigl [}{\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}{\bigr ]}}$, respectively. The group composition operation is represented as matrix multiplication. Larger signed permutation matrices represent in the same way the hyperoctahedral groups, the groups of symmetries of higher dimensional cubes, octahedra, hypercubes, and cross polytopes. ^[4]

Subgroups

Subgroups of D4

D₄ has three subgroups of order four, one consisting of its two non-involutory elements and their square (that is, its rotations, for the group's action on a square) and two more generated by two perpendicular reflections.

Each reflection generates an order-two subgroup, and there is one more order-two subgroup generated by the central symmetry (the square of the non-involutory elements).

Normal subgroups

There are four proper non-trivial normal subgroups: The two order-four subgroups are normal, as is the group generated by the central symmetry.

This version of the Cayley table shows one of these normal subgroups, shown with a red background. In this table r means rotations, and f means flips. Because this subgroup is normal, the left coset is the same as the right coset.

Group table of D₄

	e	r1	r2	r3	fv	fh	fd	fc
e	e	r1	r2	r3	fv	fh	fd	fc
r1	r1	r2	r3	e	fc	fd	fv	fh
r2	r2	r3	e	r1	fh	fv	fc	fd
r3	r3	e	r1	r2	fd	fc	fh	fv
fv	fv	fd	fh	fc	e	r2	r1	r3
fh	fh	fc	fv	fd	r2	e	r3	r1
fd	fd	fh	fc	fv	r3	r1	e	r2
fc	fc	fv	fd	fh	r1	r3	r2	e
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.

See also

• Dihedral group of order 6

References

1. Johnston, Bernard L.; Richman, Fred (1997). Numbers and Symmetry: An Introduction to Algebra. CRC Press. p. 92. ISBN   9780849303012.
2. Cameron, Peter Jephson (1998). Introduction to Algebra. Oxford University Press. p. 100. ISBN   9780198501954.
3. Humphreys, J. F. (1996). A Course in Group Theory. Oxford University Press. p. 47. ISBN   9780198534594.
4. Estévez, Manuel; Roldán, Érika; Segerman, Henry (2023). "Surfaces in the tesseract". In Holdener, Judy; Torrence, Eve; Fong, Chamberlain; Seaton, Katherine (eds.). Proceedings of Bridges 2023: Mathematics, Art, Music, Architecture, Culture. Phoenix, Arizona: Tessellations Publishing. pp. 441–444. arXiv: 2311.06596 . ISBN   978-1-938664-45-8.

External links

• "Dihedral group D8". Groupprops, The Group Properties Wiki.