Dihedral group of order 8

caption — The dihedral group of order 8 as the symmetry group of a square

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] A square maintains its appearance under 90° rotations about its center (4-fold rotational symmetry), and under reflections across four lines passing through the center – two lines parallel to the sides and its two diagonals (reflection symmetry).^[3]

Symmetries of a square

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"); each of these is a symmetry transformation. For instance, when the square is turned 90° clockwise or reflected across its vertical axis, it still looks the same, so these two movements are symmetry transformations.

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

Every point inside the square which does not lie on any of its four axes of symmetry will be mapped to 8 distinct points by these transformations, collectively called the point's orbit. An octant of the square (the region between one horizontal or vertical axis and one diagonal axis, an isosceles right triangle) is a fundamental domain of the square's surface under these symmetry transformations, containing one point from each possible orbit. One octant is highlighted in blue in the images above, to show how each symmetry transformation maps it onto a distinct octant.

Generating the group

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

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

With composition as the operation, the set of all those movements forms a group: Applying two symmetry transformations in succession yields a symmetry transformation, so the transformations are closed under composition. The symmetries of the square can be fully described by this group's Cayley table, analogous to a multiplication table but using the group operation instead of multiplication. To draw such a table explicitly, each symmetry must be assigned some symbol. In the following table, each symmetry is represented concretely by a curved arrow (rotation), double-ended arrow (reflection), or dot (identity transformation), corresponding to the images above:

$\circ$

Each entry in this table represents the composition of two symmetries: that is, the symmetry transformation resulting from performing first the symmetry transformation for the column and then performing the symmetry transformation for the row. For instance, a reflection in the diagonal NE–SW direction () followed by a horizontal reflection () is equivalent to a three-quarters-turn rotation anticlockwise (), which can be written as an equation $\circ$ $=$ , using the convention that $x\circ y$ means applying ⁠ $x$ ⁠ after ⁠ $y$ ⁠. Thus the table entry for row and column is .

Instead of pictorial symbols, mathematicians more commonly use letters. The juxtaposition of two letters ⁠ $xy$ ⁠ represents the composition of their respective symmetries ⁠ $x\circ y$ ⁠, and an integer superscript represents the composition of the same symmetry multiple times, e.g. ⁠ $x^{3}$ ⁠ represents ⁠ $x\circ x\circ x$ ⁠ and ⁠ $x^{-1}$ ⁠ represents the inverse of ⁠ $x$ ⁠. The letter ⁠ $e$ ⁠ is commonly used for the identity element. The use of this shorthand reduces the number of distinct symbols required to represent group elements.

Any element which can be written as the composition of a particular set of elements is said to be generated by those elements, for example the element ⁠ $x^{2}yx=x\circ x\circ y\circ x$ ⁠ is generated by ⁠ $x$ ⁠ and ⁠ $y$ ⁠; any set of elements which generates the whole group is called a generating set of the group. Every dihedral group can be generated from two elements, so only three elements of the group (those two and the identity) need be assigned symbols, and the other elements can be expressed by combining them.

The symmetries of the square can be generated in two basic ways: using one diagonal reflection and one horizontal or vertical reflection, or using one quarter-turn rotation and one reflection. For example, letting the letter ⁠ $r$ ⁠ represent a quarter-turn rotation anticlockwise (), letting the letter ⁠ $v$ ⁠ represent a vertical reflection (), and letting the letter ⁠ $d$ ⁠ represent a diagonal NW–SE reflection (), any pair of these elements generates the group, and the rest of the group elements can be written in terms of these. Concretely, each symmetry could be written as any of the following (the various expressions in each row represent the same symmetry):

⁠ $e$ ⁠	⁠ $e$ ⁠	⁠ $e$ ⁠
⁠ $r$ ⁠	⁠ $r$ ⁠	⁠ $dv$ ⁠
⁠ $r^{2}$ ⁠	⁠ $r^{2}$ ⁠	⁠ $(dv)^{2}$ ⁠
⁠ $r^{3}$ ⁠	⁠ $r^{3}$ ⁠	⁠ $vd$ ⁠
⁠ $rv$ ⁠	⁠ $d$ ⁠	⁠ $d$ ⁠
⁠ $r^{2}v$ ⁠	⁠ $rd$ ⁠	⁠ $dvd$ ⁠
⁠ $r^{3}v$ ⁠	⁠ $r^{2}d$ ⁠	⁠ $vdv$ ⁠
⁠ $v$ ⁠	⁠ $r^{3}d$ ⁠	⁠ $v$ ⁠

Even after choosing a pair of generators, these expressions are not unique, e.g. ⁠ $r^{3}v=vr$ ⁠. If for example we use the first column above to name the group elements, the Cayley table could be written as:

$\circ$	⁠ $e$ ⁠	⁠ $r$ ⁠	⁠ $r^{2}$ ⁠	⁠ $r^{3}$ ⁠	⁠ $rv$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $v$ ⁠
⁠ $e$ ⁠	⁠ $e$ ⁠	⁠ $r$ ⁠	⁠ $r^{2}$ ⁠	⁠ $r^{3}$ ⁠	⁠ $rv$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $v$ ⁠
⁠ $r$ ⁠	⁠ $r$ ⁠	⁠ $r^{2}$ ⁠	⁠ $r^{3}$ ⁠	⁠ $e$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $v$ ⁠	⁠ $rv$ ⁠
⁠ $r^{2}$ ⁠	⁠ $r^{2}$ ⁠	⁠ $r^{3}$ ⁠	⁠ $e$ ⁠	⁠ $r$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $v$ ⁠	⁠ $rv$ ⁠	⁠ $r^{2}v$ ⁠
⁠ $r^{3}$ ⁠	⁠ $r^{3}$ ⁠	⁠ $e$ ⁠	⁠ $r$ ⁠	⁠ $r^{2}$ ⁠	⁠ $v$ ⁠	⁠ $rv$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $r^{3}v$ ⁠
⁠ $rv$ ⁠	⁠ $rv$ ⁠	⁠ $v$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $e$ ⁠	⁠ $r^{3}$ ⁠	⁠ $r^{2}$ ⁠	⁠ $r$ ⁠
⁠ $r^{2}v$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $rv$ ⁠	⁠ $v$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $r$ ⁠	⁠ $e$ ⁠	⁠ $r^{3}$ ⁠	⁠ $r^{2}$ ⁠
⁠ $r^{3}v$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $rv$ ⁠	⁠ $v$ ⁠	⁠ $r^{2}$ ⁠	⁠ $r$ ⁠	⁠ $e$ ⁠	⁠ $r^{3}$ ⁠
⁠ $v$ ⁠	⁠ $v$ ⁠	⁠ $r^{3}v$ ⁠	⁠ $r^{2}v$ ⁠	⁠ $rv$ ⁠	⁠ $r^{3}$ ⁠	⁠ $r^{2}$ ⁠	⁠ $r$ ⁠	⁠ $e$ ⁠

Two elements ⁠ $x$ ⁠ and ⁠ $y$ ⁠ are conjugate to each-other if there is an element ⁠ $g$ ⁠ such that ⁠ $x=gyg^{-1}$ ⁠. The set of all elements which are pairwise conjugate is called a conjugacy class. These classes are important because two elements which are conjugate to each-other cannot be distinguished using only the group structure. The dihedral group of order 8 has 5 conjugacy classes: $\{$ $\}$ , $\{$ $\}$ , $\{$ $,$ $\}$ , $\{$ $,$ $\}$ , $\{$ $,$ $\}$ .

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 $\circ$ 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.^[4]

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

Symmetry transformations of an axis-aligned square centered at the origin are linear transformations, so can be represented by matrices acting on the plane by multiplication on column vectors of coordinates ${\bigl [}{\begin{smallmatrix}x\\y\end{smallmatrix}}{\bigr ]}$ . Specifically, these transformations correspond to signed permutation matrices.

The identity transformation () is represented by the identity matrix ${\bigl [}{\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}{\bigr ]}$ . A horizontal reflection () is represented by the matrix ${\bigl [}{\begin{smallmatrix}-1&0\\0&1\end{smallmatrix}}{\bigr ]}$ which negates the ⁠ $x$ ⁠ coordinate, while a vertical reflection () is represented by the matrix ${\bigl [}{\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}}{\bigr ]}$ which negates the ⁠ $y$ ⁠ coordinate. The NW–SE diagonal reflection () is represented by the matrix ${\bigl [}{\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}{\bigr ]}$ which exchanges the two coordinates, while the NE–SW diagonal reflection () is represented by its negative ${\bigl [}{\begin{smallmatrix}0&-1\\-1&0\end{smallmatrix}}{\bigr ]}$ . A rotation anticlockwise by a quarter turn () is represented by the matrix ${\bigl [}{\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}{\bigr ]}$ , and a rotation by three quarters of a turn () by its negative ${\bigl [}{\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}{\bigr ]}$ . Finally, a rotation by a half-turn (), equivalent to a reflection across the center point, is represented by the matrix ${\bigl [}{\begin{smallmatrix}-1&0\\0&-1\end{smallmatrix}}{\bigr ]}$ which negates both coordinates. 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.^[5]

Subgroups

A subgroup of a group is a subset of the elements which also form a group. $D 4$ has three subgroups of order four: one consisting of the four rotations of the square by quarter turns ( $\{$ $,$ $,$ $,$ $\}$ ) which is a cyclic group of order 4, and two others generated by pairs of perpendicular reflections ( $\{$ $,$ $,$ $,$ $\}$ and $\{$ $,$ $,$ $,$ $\}$ ) each of which is a dihedral group $D 2$ (also known as the Klein four-group). Each of the four reflections generates a cyclic subgroup of order 2 ( $\{$ $,$ $\}$ , $\{$ $,$ $\}$ , $\{$ $,$ $\}$ , and $\{$ $,$ $\}$ ). One final cyclic subgroup of order 2 consists of the rotations by half-turns ( $\{$ $,$ $\}$ ).

Normal subgroups

A subgroup is called normal if it contains all of the conjugates of its elements. Both of the groups of rotations, as well as the dihedral groups of order 4, are normal subgroups. The four subgroups generated by a single reflection are not normal.

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	f_v	f_h	f_d	f_c
e	e	r₁	r₂	r₃	f_v	f_h	f_d	f_c
r₁	r₁	r₂	r₃	e	f_c	f_d	f_v	f_h
r₂	r₂	r₃	e	r₁	f_h	f_v	f_c	f_d
r₃	r₃	e	r₁	r₂	f_d	f_c	f_h	f_v
f_v	f_v	f_d	f_h	f_c	e	r₂	r₁	r₃
f_h	f_h	f_c	f_v	f_d	r₂	e	r₃	r₁
f_d	f_d	f_h	f_c	f_v	r₃	r₁	e	r₂
f_c	f_c	f_v	f_d	f_h	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.

References

↑ Johnston, Bernard L.; Richman, Fred (1997). Numbers and Symmetry: An Introduction to Algebra. CRC Press. p. 92. ISBN 9780849303012.
↑ Cameron, Peter Jephson (1998). Introduction to Algebra. Oxford University Press. p. 100. ISBN 9780198501954.
↑ Krantz, Steven G. (2011). The Proof is in the Pudding: The Changing Nature of Mathematical Proof. Springer Science & Business Media. p. 160. ISBN 978-0-387-48744-1.
↑ Humphreys, J. F. (1996). A Course in Group Theory. Oxford University Press. p. 47. ISBN 9780198534594.
↑ 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.
Wikiversity:Symmetric group S4

[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] Krantz, Steven G. (2011). The Proof is in the Pudding: The Changing Nature of Mathematical Proof. Springer Science & Business Media. p. 160. ISBN 978-0-387-48744-1.

[4] Humphreys, J. F. (1996). A Course in Group Theory. Oxford University Press. p. 47. ISBN 9780198534594.

[5] 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.

[1]

[2]

[3]

[4]

[5]