TheInfoList

OR: In mathematics, a dihedral group is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
, which includes
rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
and reflections. Dihedral groups are among the simplest examples of
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
s, and they play an important role in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, and
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, ...
. The notation for the dihedral group differs in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
and
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
. In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, or refers to the symmetries of the -gon, a group of order . In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, refers to this same dihedral group. This article uses the geometric convention, .

# Definition

## Elements A regular polygon with $n$ sides has $2n$ different symmetries: $n$
rotational symmetries Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
and $n$
reflection symmetries In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D the ...
. Usually, we take $n \ge 3$ here. The associated
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s and reflections make up the dihedral group $\mathrm_n$. If $n$ is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If $n$ is even, there are $n/2$ axes of symmetry connecting the midpoints of opposite sides and $n/2$ axes of symmetry connecting opposite vertices. In either case, there are $n$ axes of symmetry and $2n$ elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes. The following picture shows the effect of the sixteen elements of $\mathrm_8$ on a
stop sign A stop sign is a traffic sign designed to notify drivers that they must come to a complete stop and make sure the intersection is safely clear of vehicles and pedestrians before continuing past the sign. In many countries, the sign is a red oct ...
: The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left.

## Group structure

As with any geometric object, the
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
. The following
Cayley table Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplic ...
shows the effect of composition in the group D3 (the symmetries of an
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
). r0 denotes the identity; r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, and s0, s1 and s2 denote reflections across the three lines shown in the adjacent picture. For example, , because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group D''n'' has elements r0, ..., r''n''−1 and s0, ..., s''n''−1, with composition given by the following formulae: :$\mathrm_i\,\mathrm_j = \mathrm_, \quad \mathrm_i\,\mathrm_j = \mathrm_, \quad \mathrm_i\,\mathrm_j = \mathrm_, \quad \mathrm_i\,\mathrm_j = \mathrm_.$ In all cases, addition and subtraction of subscripts are to be performed using
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
with modulus ''n''.

## Matrix representation If we center the regular polygon at the origin, then elements of the dihedral group act as
linear transformations In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pres ...
of the plane. This lets us represent elements of D''n'' as
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, with composition being
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
. This is an example of a (2-dimensional)
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
. For example, the elements of the group D4 can be represented by the following eight matrices: : In general, the matrices for elements of D''n'' have the following form: :$\begin \mathrm_k & = \begin \cos \frac & -\sin \frac \\ \sin \frac & \cos \frac \end\ \ \text \\ \mathrm_k & = \begin \cos \frac & \sin \frac \\ \sin \frac & -\cos \frac \end . \end$ r''k'' is a rotation matrix, expressing a counterclockwise rotation through an angle of . s''k'' is a reflection across a line that makes an angle of with the ''x''-axis.

## Other definitions

Further equivalent definitions of are:

# Small dihedral groups is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word iso ...
to , the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
of order 2. is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word iso ...
to , the Klein four-group. and are exceptional in that: * and are the only abelian dihedral groups. Otherwise, is non-abelian. * is a subgroup of the symmetric group for . Since for or , for these values, is too large to be a subgroup. * The inner automorphism group of is trivial, whereas for other even values of , this is . The cycle graphs of dihedral groups consist of an ''n''-element cycle and ''n'' 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
.

# The dihedral group as symmetry group in 2D and rotation group in 3D

An example of abstract group , and a common way to visualize it, is the group of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. consists of
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s of multiples of about the origin, and reflections across lines through the origin, making angles of multiples of with each other. This is the symmetry group of a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
with sides (for ; this extends to the cases and where we have a plane with respectively a point offset from the "center" of the "1-gon" and a "2-gon" or line segment). is generated by a rotation of
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
and a reflection of order 2 such that :$\mathrm = \mathrm^ \,$ In geometric terms: in the mirror a rotation looks like an inverse rotation. In terms of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the f ...
: multiplication by $e^$ and
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. In matrix form, by setting : and defining $\mathrm_j = \mathrm_1^j$ and $\mathrm_j = \mathrm_j \, \mathrm_0$ for $j \in \$ we can write the product rules for D''n'' as :$\begin \mathrm_j \, \mathrm_k &= \mathrm_ \\ \mathrm_j \, \mathrm_k &= \mathrm_ \\ \mathrm_j \, \mathrm_k &= \mathrm_ \\ \mathrm_j \, \mathrm_k &= \mathrm_ \end$ (Compare coordinate rotations and reflections.) The dihedral group D2 is generated by the rotation r of 180 degrees, and the reflection s across the ''x''-axis. The elements of D2 can then be represented as , where e is the identity or null transformation and rs is the reflection across the ''y''-axis. D2 is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word iso ...
to the Klein four-group. For ''n'' > 2 the operations of rotation and reflection in general do not
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
and D''n'' is not abelian; for example, in D4, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees. Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups. The elements of can be written as , , , ... , , , , , ... , . The first listed elements are rotations and the remaining elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection. So far, we have considered to be a subgroup of , i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation is also used for a subgroup of
SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin ...
which is also of abstract group type : the proper symmetry group of a ''regular polygon embedded in three-dimensional space'' (if ''n'' ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore, it is also called a ''dihedron'' (Greek: solid with two faces), which explains the name ''dihedral group'' (in analogy to ''tetrahedral'', ''octahedral'' and ''icosahedral group'', referring to the proper symmetry groups of a regular tetrahedron, octahedron, and icosahedron respectively).

## Examples of 2D dihedral symmetry

File:Imperial Seal of Japan.svg, 2D D16 symmetry – Imperial Seal of Japan, representing eightfold
chrysanthemum Chrysanthemums (), sometimes called mums or chrysanths, are flowering plants of the genus ''Chrysanthemum'' in the family Asteraceae. They are native to East Asia and northeastern Europe. Most species originate from East Asia and the cente ...
with sixteen
petal Petals are modified leaves that surround the reproductive parts of flowers. They are often brightly colored or unusually shaped to attract pollinators. All of the petals of a flower are collectively known as the ''corolla''. Petals are usually ...
s. File:Red Star of David.svg, 2D D6 symmetry – The Red Star of David File:Naval Jack of the Republic of China.svg, 2D D12 symmetry — The Naval Jack of the Republic of China (White Sun) File:Ashoka Chakra.svg, 2D D24 symmetry –
Ashoka Chakra Ashoka (, ; also ''Asoka''; 304 – 232 BCE), popularly known as Ashoka the Great, was the third emperor of the Maurya Empire of Indian subcontinent during to 232 BCE. His empire covered a large part of the Indian subcontinent, s ...
, as depicted on the National flag of the Republic of India.

# Properties

The properties of the dihedral groups with depend on whether is even or odd. For example, the
center Center or centre may refer to: Mathematics * Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentr ...
of consists only of the identity if ''n'' is odd, but if ''n'' is even the center has two elements, namely the identity and the element r''n''/2 (with D''n'' as a subgroup of O(2), this is inversion; since it is scalar multiplication by −1, it is clear that it commutes with any linear transformation). In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones. For ''n'' twice an odd number, the abstract group is isomorphic with the direct product of and . Generally, if ''m''
divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
''n'', then has ''n''/''m'' subgroups of type , and one subgroup $\mathbb$''m''. Therefore, the total number of subgroups of (''n'' ≥ 1), is equal to ''d''(''n'') + σ(''n''), where ''d''(''n'') is the number of positive
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of ''n'' and ''σ''(''n'') is the sum of the positive divisors of ''n''. See list of small groups for the cases ''n'' ≤ 8. The dihedral group of order 8 (D4) is the smallest example of a group that is not a T-group. Any of its two Klein four-group subgroups (which are normal in D4) has as normal subgroup order-2 subgroups generated by a reflection (flip) in D4, but these subgroups are not normal in D4.

## Conjugacy classes of reflections

All the reflections are
conjugate Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
to each other whenever ''n'' is odd, but they fall into two conjugacy classes if ''n'' is even. If we think of the isometries of a regular ''n''-gon: for odd ''n'' there are rotations in the group between every pair of mirrors, while for even ''n'' only half of the mirrors can be reached from one by these rotations. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon there are two sets of axes, each corresponding to a conjugacy class: those that pass through two vertices and those that pass through two sides. Algebraically, this is an instance of the conjugate Sylow theorem (for ''n'' odd): for ''n'' odd, each reflection, together with the identity, form a subgroup of order 2, which is a Sylow 2-subgroup ( is the maximum power of 2 dividing ), while for ''n'' even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group. For ''n'' even there is instead an
outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...
interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism).

# Automorphism group

The automorphism group of is isomorphic to the holomorph of $\mathbb$/''n''$\mathbb$, i.e., to and has order ''nϕ''(''n''), where ''ϕ'' is Euler's totient function, the number of ''k'' in coprime to ''n''. It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by ''k''(2''π''/''n''), for ''k''
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to ''n''); which automorphisms are inner and outer depends on the parity of ''n''. * For ''n'' odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for ''n'' even, the rotation by 180° (reflection through the origin) is the non-trivial element of the center. * Thus for ''n'' odd, the inner automorphism group has order 2''n'', and for ''n'' even (other than ) the inner automorphism group has order ''n''. * For ''n'' odd, all reflections are conjugate; for ''n'' even, they fall into two classes (those through two vertices and those through two faces), related by an outer automorphism, which can be represented by rotation by ''π''/''n'' (half the minimal rotation). * The rotations are a normal subgroup; conjugation by a reflection changes the sign (direction) of the rotation, but otherwise leaves them unchanged. Thus automorphisms that multiply angles by ''k'' (coprime to ''n'') are outer unless .

## Examples of automorphism groups

has 18
inner automorphism In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
s. As 2D isometry group D9, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are all automorphisms. As abstract group there are in addition to these, 36
outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...
s; e.g., multiplying angles of rotation by 2. has 10 inner automorphisms. As 2D isometry group D10, the group has mirrors at 18° intervals. The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect to the inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20 more outer automorphisms; e.g., multiplying rotations by 3. Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo ''n'' for ''n'' = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order). The only values of ''n'' for which ''φ''(''n'') = 2 are 3, 4, and 6, and consequently, there are only three dihedral groups that are isomorphic to their own automorphism groups, namely (order 6), (order 8), and (order 12).

## Inner automorphism group

The inner automorphism group of is isomorphic to: * if ''n'' is odd; * if is even (for , ).

# Generalizations

There are several important generalizations of the dihedral groups: * The
infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1' ...
is an infinite group with algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. * The
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
O(2), ''i.e.,'' the symmetry group of the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, also has similar properties to the dihedral groups. * The family of generalized dihedral groups includes both of the examples above, as well as many other groups. * The quasidihedral groups are family of finite groups with similar properties to the dihedral groups.

* Coordinate rotations and reflections * Cycle index of the dihedral group * Dicyclic group *
Dihedral group of order 6 In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abel ...
* Dihedral group of order 8 * Dihedral symmetry groups in 3D * Dihedral symmetry in three dimensions