In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or General properties These properties apply to both convex and star regular polygons Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties

See also: Dihedral symmetry in three dimensions. 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 Dih

Contents 1 Notation 2 Definition 2.1 Elements 2.2 Group structure 2.3 Matrix representation 3 Small dihedral groups 4 The dihedral group as symmetry group in 2D and rotation group in 3D 5 Equivalent definitions and properties 6 Examples of automorphism groups 7 Infinite dihedral group 8 Generalized dihedral group 9 Topology 10 See also 11 External links

Notation

There are two competing notations for the dihedral group associated to a polygon with n sides. See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering In geometry the group is denoted D_n, while in algebra the same group is denoted by D_2n to indicate the number of elements. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules

In this article, D_n (and sometimes Dih_n) refers to the symmetries of a regular polygon with n sides.

Definition

Elements

A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect The associated rotations and reflections make up the dihedral group D_n. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. The following picture shows the effect of the sixteen elements of D₈ on a stop sign:

The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections. A stop sign is a Traffic sign, usually erected at Road junctions that instructs drivers to stop and then to proceed only if the way ahead is clear

Group structure

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. In Mathematics, a composite function represents the application of one function to the results of another This operation gives the symmetries of a polygon the algebraic structure of a finite group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

The following Cayley table shows the effect of composition in the group D₃ (the symmetries of an equilateral triangle). A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a Properties The area of an equilateral triangle with sides of length a\\! R₀ denotes the identity; R₁ and R₂ denote counterclockwise rotations by 120 and 240 degrees; and S₀, S₁, and S₂ denote reflections across the three lines shown in the picture to the right.

	R0	R1	R2	S0	S1	S2
R0	R0	R1	R2	S0	S1	S2
R1	R1	R2	R0	S1	S2	S0
R2	R2	R0	R1	S2	S0	S1
S0	S0	S2	S1	R0	R2	R1
S1	S1	S0	S2	R1	R0	R2
S2	S2	S1	S0	R2	R1	R0

For example, S₂S₁ = R₁ because the reflection S₁ followed by the reflection S₂ results in a 120-degree rotation. (This is the normal backwards order for composition. In Mathematics, a composite function represents the application of one function to the results of another ) Note that the composition operation is not commutative. In Mathematics, commutativity is the ability to change the order of something without changing the end result

In general, the group D_n has elements R₀,. . . ,R_n−1 and S₀,. . . ,S_n−1, with composition given by the following formulae:

In all cases, addition and subtraction of subscripts should be performed using modular arithmetic with modulus n. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

Matrix representation

If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane This lets us represent elements of D_n as matrices, with composition being matrix multiplication. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix 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 Linear transformations of

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:

    and    

The first matrix is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk ⁄ n. In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 The second matrix is a reflection across a line that makes an angle of πk ⁄ n with the x-axis.

Small dihedral groups

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. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 Both are exceptional within the series:

• they are abelian; for all other values of n the group Dih_n is not abelian
• they are not subgroups of the symmetric group S_n, corresponding to the fact that 2n > n ! for these n. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. In Group theory, a sub-field of Abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful 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, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

Dih1	Dih2	Dih3	Dih4	Dih5	Dih6	Dih7

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. The emblems of the International Red Cross and Red Crescent Movement, under the Geneva Conventions, are to be placed on Humanitarian and medical vehicles 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 These groups form one of the two series of discrete point groups in two dimensions. 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 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. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. 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). The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is General properties These properties apply to both convex and star regular polygons

Dihedral group D_n is generated by a rotation r of order n and a reflection f of order 2 such that

frf = r ^{− 1} (in geometric terms: in the mirror a rotation looks like an inverse rotation)

In matrix form, an anti-clockwise rotation and a reflection in the x-axis are given by

(in terms of complex numbers: multiplication by and complex conjugation). In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.

By setting

and defining and for we can write the product rules for D_n as

(Compare coordinate rotations and reflections. In Geometry, 2D Coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another )

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. 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 In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2

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:

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. In Mathematics, commutativity is the ability to change the order of something without changing the end result An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or

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. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n 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). This article is about rotations in three-dimensional Euclidean space The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is 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). 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 In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ

Equivalent definitions and properties

Further equivalent definitions of Dih_n are:

• The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3). In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects
• The group with presentation

or

(Indeed the only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclic groups)

From the second presentation follows that Dih_n belongs to the class of coxeter groups. In Mathematics, one method of defining a group is by a presentation. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries

• The semidirect product of cyclic groups Z_n and Z₂, with Z₂ acting on Z_n by inversion (thus, Dih_n always has a normal subgroup isomorphic to Z_n ):

is isomorphic to Dih_n if φ(0) is the identity and φ(1) is inversion. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.

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 In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

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 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 Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in

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 objectsalready 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. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class 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.

If m divides n, then Dih_n has n / m subgroups of type Dih_m, and one subgroup Z_m. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of 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. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without See List of small groups for the cases n ≤ 8. 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. In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G 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. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner g. multiplying angles of rotation by 2.

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. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to In Modular arithmetic the set of Congruence classes Relatively prime to the modulus n form a group under multiplication called the multiplicative 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 general, the automorphism group of Dih_n is isomorphic to the affine group Aff(Z/nZ). In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible

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_∞. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness It has presentations

and is isomorphic to a semidirect product of Z and Z₂, and to the free product Z₂ * Z₂. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Abstract algebra, the free product of groups constructs a group from two or more given ones 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 Mathematics, the isometry group of a Metric space is the set of all isometries from the metric space onto itself with the Function composition A one-dimensional symmetry group is a mathematical group that describe symmetries in one dimension

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. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can I. e. , with φ(0) the identity and φ(1) inversion.

Thus we get:

(h₁, 0) * (h₂, t₂) = (h₁ + h₂, t₂)

(h₁, 1) * (h₂, t₂) = (h₁ - h₂, 1 + t₂)

for all h₁, h₂ in H and t₂ in Z₂.

(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, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH

The conjugacy classes are:

• the sets {(h,0 ), (-h,0 )}
• the sets {(h + k + k, 1) | k in H }

Thus for every subgroup M of H, the corresponding set of elements (m,0) is also a normal subgroup. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class We have:

Dih(H) / M = Dih ( H / M )

Examples:

• Dih_n = Dih(Z_n)
  • For even n there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dih_{n / 2}. As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has (the rotations of both are the same). However, they are isomorphic as abstract groups.
  • For odd n there is only one set {(h + k + k, 1) | k in H }
• Dih_∞ = Dih(Z); there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dih_∞. As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between (the translations of both are the same: by even numbers). However, they are isomorphic as abstract groups.
• Dih(S¹), or orthogonal group O(2,R), or O(2): the isometry group of a circle, or equivalently, the group of isometries in 2D that keep the origin fixed. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The rotations form the circle group S¹, or equivalently SO(2,R), also written SO(2), and R/Z ; it is also the multiplicative group of complex numbers of absolute value 1. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In the latter case one of the reflections (generating the others) is complex conjugation. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. There are no proper normal subgroups with reflections. The discrete normal subgroups are cyclic groups of order n for all positive integers n. The quotient groups are isomorphic with the same group Dih(S¹).
• Dih(Rⁿ ): the group of isometries of Rⁿ consisting of all translations and inversion in all points; for n = 1 this is the Euclidean group E(1); for n > 1 the group Dih(Rⁿ ) is a proper subgroup of E(n ), i. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional A one-dimensional symmetry group is a mathematical group that describe symmetries in one dimension e. it does not contain all isometries.
• H can be any subgroup of Rⁿ, e. g. a discrete subgroup; in that case, if it extends in n directions it is a lattice. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of
  • Discrete subgroups of Dih(R² ) which contain translations in one direction are of frieze group type and 22. A frieze group is a mathematical concept to classify designs on Two-dimensional surfaces which are repetitive in one direction based on the symmetries in the pattern
  • Discrete subgroups of Dih(R² ) which contain translations in two directions are of wallpaper group type p1 and p2. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the
  • Discrete subgroups of Dih(R³ ) which contain translations in three directions are space groups of the triclinic crystal system. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions

Dih(H) is Abelian, with the semidirect product a direct product, if and only if all elements of H are their own inverse:

• Dih(Z₁) = Dih₁ = Z₂
• Dih(Z₂) = Dih₂ = Z₂ × Z₂ (Klein four-group)
• Dih(Dih₂) = Dih₂ × Z₂ = Z₂ × Z₂ × Z₂

etc. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2

Topology

Dih(Rⁿ ) and its dihedral subgroups are disconnected topological groups. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Dih(Rⁿ ) consists of two connected components: the identity component isomorphic to Rⁿ, and the component with the reflections. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity Similarly O(2) consists of two connected components: the identity component isomorphic to the circle group, and the component with the reflections.

For the group Dih_∞ we can distinguish two cases:

• Dih_∞ as the isometry group of Z
• Dih_∞ as a 2-dimensional isometry group generated by a rotation by an irrational number of turns, and a reflection

Both topological groups are totally disconnected, but in the first case the (singleton) components are open, while in the second case they are not. In Topology and related branches of Mathematics, a totally disconnected space is a Topological space which is maximally disconnected in the sense that Also, the first topological group is a closed subgroup of Dih(R) but the second is not a closed subgroup of O(2).

See also

• quasidihedral group
• dicyclic group
• coordinate rotations and reflections
• dihedral group of order 6
• dihedral group of order 8
• dihedral symmetry in three dimensions
• dihedral symmetry groups in 3D

External links

• Dihedral Group n of Order 2n by Shawn Dudzik, The Wolfram Demonstrations Project. In Mathematics, the quasi-dihedral groups and semi-dihedral groups are non-abelian groups of order a power of 2 In Group theory, a dicyclic group is a member of a class of groups Dic n ( n > 1 a Non-abelian group of order 4 n In Geometry, 2D Coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another The smallest Non-abelian group has 6 elements It is a Dihedral group with notation D 3 or D 6 Some elementary examples of groups in Mathematics are given on Group (mathematics. 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 Dih In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group

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