Dihedral group of order 8

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, {{tmath|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 {{tmath|b}}. The movement that does nothing is denoted by {{tmath|e}}.

{{multiple image

| width = 180

| perrow = 4

| align = center

| image_style=border:none

| image1 = Square symmetry – I.png

| caption1 = The square's initial position
(the identity transformation)

| image2 = Square symmetry – R1.png

| caption2 = Rotation by 90° anticlockwise

| image3 = Square symmetry – R2.png

| caption3 = Rotation by 180°

| image4 = Square symmetry – R3.png

| caption4 = Rotation by 270°

| image5 = Square symmetry – D1.png

| caption5 = Diagonal NW–SE reflection

| image6 = Square symmetry – H.png

| caption6 = Horizontal reflection

| image7 = Square symmetry – D2.png

| caption7 = Diagonal NE–SW reflection

| image8 = Square symmetry – V.png

| caption8 = Vertical reflection

}}

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 $\backslash circ$ 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 $\backslash circ$ b is the same as b $\backslash circ$ a³. Also, a² $\backslash circ$ b is a vertical flip and is equal to b $\backslash circ$ 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.

File:Dih4 cycle graph.svg of Dih₄
a is the clockwise rotation
and b the horizontal reflection.]]

File:Dih 4 Cayley Graph; generators a, b.svg of Dih₄]]

File:Dih 4 Cayley Graph; generators b, c.svg

This group of order 8 has the following Cayley table:

For any two elements in the group, the table records what their composition is. Here we wrote "a³b" as a shorthand for a³ $\backslash circ$ b. This group has 5 conjugacy classes, they are $\backslash \{e\backslash \},\; \backslash \{a^2\backslash \},\; \backslash \{b,ba^2\backslash \},\backslash \{ba,ba^3\backslash \},\; \backslash text\{\; and\; \}\; \backslash \{a,a^3\backslash \}$.

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 $\backslash circ$ 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.{{cite book|first=J. F. |last=Humphreys |title=A Course in Group Theory |year=1996 |isbn=9780198534594 |publisher=Oxford University Press |page=47}}

File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,2,2,4,2); subgroup of S4.svg 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).

{{-}}

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 $\backslash bigl[\backslash begin\{smallmatrix\}x\backslash \backslash y\backslash end\{smallmatrix\}\backslash bigr]$. The identity transformation is represented by the identity matrix .

Reflections across a horizontal and vertical axis are represented by the two matrices and , respectively, and the two diagonal reflections are represented by the matrices and .

Rotations clockwise by 90°, 180°, and 270° are represented by the matrices , , and , 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.{{cite conference

| last1 = Estévez | first1 = Manuel

| last2 = Roldán | first2 = Érika

| last3 = Segerman | first3 = Henry | author3-link = Henry Segerman

| editor1-last = Holdener | editor1-first = Judy | editor1-link = Judy Holdener

| editor2-last = Torrence | editor2-first = Eve | editor2-link = Eve Torrence

| editor3-last = Fong | editor3-first = Chamberlain

| editor4-last = Seaton | editor4-first = Katherine

| arxiv = 2311.06596

| contribution = Surfaces in the tesseract

| contribution-url = https://archive.bridgesmathart.org/2023/bridges2023-441.html

| isbn = 978-1-938664-45-8

| location = Phoenix, Arizona

| pages = 441–444

| publisher = Tessellations Publishing

| title = Proceedings of Bridges 2023: Mathematics, Art, Music, Architecture, Culture

| year = 2023}}

File:Dih4 subgroups.svg

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).

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.

:

{{clear}}

• Dihedral group of order 6

{{reflist}}

• {{cite web|url=https://groupprops.subwiki.org/wiki/Dihedral_group:D8 |title=Dihedral group D8 |website=Groupprops, The Group Properties Wiki}}

Category:Finite groups