NationMaster - Encyclopedia: Dihedral group

In mathematics, the dihedral group of order 2n is a certain abstract group for which here the notation Dih_n is used. For the isometry group in 2D of this abstract type, and one of them in 3D, the notation D_n is used. Often the notation D_n is also used for the abstract group, and sometimes D_2n. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... The symbols of the Movement - The Red Cross and the Red Crescent emblems at the museum in Geneva. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...

For n = 1 we have Dih₁. This notation is rarely used except in the framework of the series, because it is equal to Z₂. For n = 2 we have Dih₂, the Klein four-group. Both are exceptional within the series: This article is about the mathematical group. ...

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 stand for 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, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In group theory a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. ...

Image File history File links GroupDiagramMiniC2. ... Cycle diagram for group D4 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group D6 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group D8 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group D10 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group D12 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group D14 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ...

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

An example of abstract group Dih_n, and a common way to visualize it, is the group D_n of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. D_n consists of n rotations of multiples of 360°/n about the origin, and reflections across n lines through the origin, making angles of multiples of 180°/n with each other. This is the symmetry group of a regular polygon with n sides (for n ≥3, and also for the degenerate case n = 2, where we have a line segment in the plane). Image File history File links Red_Star_of_David. ... Image File history File links Red_Star_of_David. ... The symbols of the Movement - The Red Cross and the Red Crescent emblems at the museum in Geneva. ... In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. ... In geometry, a point group in two dimensions is an isometry group in two dimensions that leaves the origin fixed, or correspondingly, an isometry group of a circle. ... Rotation of a planar figure around a point Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation. ... The word reflection (also spelt reflexion in British English) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ... The symmetry group of an object (e. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

Dihedral group D_n is generated by a rotation r of order n and a reflection f of order 2 such that In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

In matrix form, an anti-clockwise rotation and a reflection in the x-axis are given by For the square matrix section, see square matrix. ...

(in terms of complex numbers: multiplication by $e^{2pi over n}$ and complex conjugation). The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

and defining $r_j = r_0^j$ and $f_j = r_j , f_0$ for $j in {1,ldots,n-1}$ we can write the product rules for

D n

(Compare coordinate rotations and reflections.) In geometry, coordinate rotations and reflections are two kinds of isometry which are related to each to other. ...

The dihedral group D₂ is generated by the rotation r of 180 degrees, and the reflection f across the x-axis. The elements of D₂ can then be represented as {e, r, f, rf}, where e is the identity or null transformation and rf is the reflection across the y-axis.

D₂ is isomorphic to the Klein four-group. diagram of the D4, dihedral group; made by me for wiki File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... diagram of the D4, dihedral group; made by me for wiki File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... This article is about the mathematical group. ...

If the order of D_n is greater than 4, the operations of rotation and reflection in general do not commute 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: In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an abelian group is a commutative group, i. ...

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. A diagram showing D8 is non-abelian; generated for wiki by me File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A diagram showing D8 is non-abelian; generated for wiki by me File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Square with symmetry group D<sub>4</sub> Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, an abelian group is a commutative group, i. ...

The 2n elements of D_n can be written as e, r, r²,...,rⁿ⁻¹, f, r f, r² f,...,rⁿ⁻¹ f. The first n listed elements are rotations and the remaining n 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 D_n to be a subgroup of O(2), i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation D_n is also used for a subgroup of SO(3) which is also of abstract group type Dih_n: 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). In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... The symmetry group of an object (e. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... An icosahedron [ËŒaÄ±kÉ™sÉ™hiËdrÉ™n] noun (plural: -drons, -dra [-drÉ™]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ...

See also dihedral symmetry. This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n â‰¥ 2 ). See also point groups in two dimensions. ...

Equivalent definitions and properties

Z_n

_φ Z₂ is isomorphic to Dih_n if φ(0) is the identity and φ(1) is inversion. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, one method of defining a group is by a presentation. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

If we consider Dih_n (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see that Dih_n is a subgroup of the symmetric group S_n. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

The properties of the dihedral groups Dih_n with n ≥ 3 depend on whether n is even or odd. For example, the center of Dih_n consists only of the identity if n is odd, but contains the element r^{n / 2} if n is even (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 abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ... In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ...

For odd n, abstract group Dih_2n is isomorphic with the direct product of Dih_n and Z₂. In mathematics, one can often define a direct product of objects already known, giving a new one. ...

In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.

All the reflections are conjugate to each other in case 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. In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

If m divides n, then Dih_n has n / m subgroups of type Dih_m, and one subgroup Z_m. Therefore the total number of subgroups of Dih_n (n ≥ 1), is equal to d (n) + σ (n), where d (n) is the number of positive divisors of n and σ (n) is the sum of the positive divisors of n. See List of small groups for the cases n ≤ 8. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

Examples of automorphism groups

Dih₉ has 18 inner automorphisms. 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 automorphisms, e.g. multiplying angles of rotation by 2. In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ...

Dih₁₀ 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). In number theory, the totient Ï†(n) of a positive integer n is defined to be the number of positive integers less than n and coprime to n. ... In mathematics, the multiplicative group of integers modulo n is the group defined by multiplication of the units (that is, the numbers relatively prime to ) in the ring for a given integer . ...

Infinite dihedral group

In addition to the finite dihedral groups, there is the infinite dihedral group Dih_∞. Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that rⁿ is the identity, and we have a finite dihedral group of order 2n. If the rotation is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called Dih_∞. It has presentations Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...

and is isomorphic to a semidirect product of Z and Z₂, and to the free product Z₂ * Z₂. It is the automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z (see also symmetry groups in one dimension). In abstract algebra, the free product of groups constructs a group from two or more given ones. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... A pattern in 1D can be represented as a function f(x) for, say, the color at position x. ...

Generalized dihedral group

For any abelian group H, the generalized dihedral group of H, written Dih(H), is the semidirect product of H and Z₂, with Z₂ acting on H by inverting elements. I.e., Dih(H) = H

_φ Z₂ with φ(0) the identity and φ(1) inversion. In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

(Writing Z₂ multiplicatively, we have (h₁, t₁) * (h₂, t₂) = (h₁ + t₁h₂, t₁t₂) .)

Note that (h, 0) * (0,1) = (h,1), i.e. first the inversion and then the operation in H. Also (0, 1) * (h, t) = (- h, 1 + t); indeed (0,1) inverts h, and toggles t between "normal" (0) and "inverted" (1) (this combined operation is its own inverse).

The subgroup of Dih(H) of elements (h, 0) is a normal subgroup of index 2, isomorphic to H, while the elements (h, 1) are all their own inverse. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

The conjugacy classes are: In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

Thus for every subgroup M of H, the corresponding set of elements (m,0) is also a normal subgroup. We have:

Dih(H) is Abelian, with the semidirect product a direct product, iff all elements of H are their own inverse: In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... A pattern in 1D can be represented as a function f(x) for, say, the color at position x. ... See lattice for other meanings of this term, both within and without mathematics. ... A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ... Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ... A space group is a mathematical symmetry group or symmetry group type of n-dimensional structures with translational symmetry in n independent directions, such as, for n = 3, a crystal. ... In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

Topology

Dih(Rⁿ ) and its dihedral subgroups are disconnected topological groups. Dih(Rⁿ ) consists of two connected components: the identity component isomorphic to Rⁿ, and the component with the reflections. Similarly O(2) consists of two connected components: the identity component isomorphic to the circle group, and the component with the reflections. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ... In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ...

Both topological groups are totally disconnected, but in the first case the (singleton) components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih(R) but the second is not a closed subgroup of O(2).

Contents

The dihedral group as symmetry group in 2D and rotation group in 3D GA_googleFillSlot("encyclopedia_square");