Dihedral group 2D D 4 symmetry - The Red Crystal symbol In mathematics , the dihedral group of order 2 n is the abstract group of which one representation is the symmetry group in 2D of a regular polygon with n sides . The group consists of n elements corresponding to rotations of the polygon , and n corresponding to reflections . Notation In this article the notation Dih n is used for the dihedral group of order 2 n as abstract group . The notations D n and D 2n are also seen . For the isometry group in 2D of this abstract group type , the notation D n is used . There are four series of isometry groups in 3D which are dihedral as abstract group . Only for one of them the notation D n is used . Small dihedral groups For n = 1 we have Dih 1 . This notation is rarely used except in the framework of the series , because it is equal to Z 2 . For n = 2 we have Dih 2 , the Klein four-group . 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 2 n > n   ! for these n . 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 . Dih 1 Dih 2 Dih 3 Dih 4 Dih 5 Dih 6 Dih 7 The dihedral group as symmetry group in 2D and rotation group in 3D 2D D 6 symmetry - The Red Star of David 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 ) . 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 r = \begin { bmatrix } \cos { 2\pi \over n } & -\sin { 2\pi \over n } \\ \sin { 2\pi \over n } & \cos { 2\pi \over n } \end { bmatrix } \qquad f = \begin { bmatrix } 1 & 0 \\ 0 & -1\end { bmatrix } ( in terms of complex numbers : multiplication by e^ { 2\pi i \over n } and complex conjugation ) . By setting r_0 = \begin { bmatrix } \cos { 2\pi \over n } & -\sin { 2\pi \over n } \\ \sin { 2\pi \over n } & \cos { 2\pi \over n } \end { bmatrix } \qquad f_0 = \begin { bmatrix } 1 & 0 \\ 0 & -1\end { bmatrix } 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 as r_j \ , r_k = r_ { ( j+k ) \mbox { mod n } } r_j \ , f_k = f_ { ( j+k ) \mbox { mod n } } f_j \ , r_k = f_ { ( j-k ) \mbox { mod n } } f_j \ , f_k = r_ { ( j-k ) \mbox { mod n } } ( Compare coordinate rotations and reflections . ) The dihedral group D 2 is generated by the rotation r of 180 degrees , and the reflection f across the x-axis . The elements of D 2 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 . x-axis is vertical D 2 is isomorphic to the Klein four-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 D 4 , a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees : x-axis is vertical 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 2 n elements of D n can be written as e , r , r 2 , ... , r n −1 , f , r f , r 2 f , ... , r n −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 ) . 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 ) . The group with presentation \langle r , f \mid r^n = 1 , f^2 = 1 , frf = r^ { -1 } \rangle or \langle x , y \mid x^2 = y^2 = ( xy ) ^n = 1 \rangle ( 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 . The semidirect product of cyclic groups Z n and Z 2 , with Z 2 acting on Z n by inversion ( thus , Dih n always has a normal subgroup isomorphic to Z n ) : Z n φ Z 2 is isomorphic to Dih n if φ ( 0 ) is the identity and φ ( 1 ) is inversion . 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 . 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 ) . For odd n , abstract group Dih 2 n is isomorphic with the direct product of Dih n and Z 2 . 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 . 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 . Examples of automorphism groups Dih 9 has 18 inner automorphisms . As 2D isometry group D 9 , 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 . Dih 10 has 10 inner automorphisms . As 2D isometry group D 10 , 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 ) . 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 r n is the identity , and we have a finite dihedral group of order 2 n . 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 \langle r , f \mid f^2 = 1 , frf = r^ { -1 } \rangle \langle x , y \mid x^2 = y^2 = 1 \rangle and is isomorphic to a semidirect product of Z and Z 2 , and to the free product Z 2 * Z 2 . 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 ) . Generalized dihedral group For any abelian group H , the generalized dihedral group of H , written Dih ( H ) , is the semidirect product of H and Z 2 , with Z 2 acting on H by inverting elements . I.e. , Dih ( H ) = H φ Z 2 with φ ( 0 ) the identity and φ ( 1 ) inversion . Thus we get : ( h 1 , 0 ) * ( h 2 , t 2 ) = ( h 1 + h 2 , t 2 ) ( h 1 , 1 ) * ( h 2 , t 2 ) = ( h 1 - h 2 , 1 + t 2 ) for all h 1 , h 2 in H and t 2 in Z 2 . ( Writing Z 2 multiplicatively , we have ( h 1 , t 1 ) * ( h 2 , t 2 ) = ( h 1 + t 1 h 2 , t 1 t 2 ) . ) 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 . 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 . 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 ( S 1 ) , 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 . The rotations form the circle group S 1 , 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 the latter case one of the reflections ( generating the others ) is complex conjugation . 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 ( S 1 ) . Dih ( R n ) : the group of isometries of R n 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 ( R n ) is a proper subgroup of E ( n ) , i.e. it does not contain all isometries . H can be any subgroup of R n , e.g. a discrete subgroup ; in that case , if it extends in n directions it is a lattice . Discrete subgroups of Dih ( R 2 ) which contain translations in one direction are of frieze group type \infty\infty and 22 \infty . Discrete subgroups of Dih ( R 2 ) which contain translations in two directions are of wallpaper group type p1 and p2 . Discrete subgroups of Dih ( R 3 ) which contain translations in three directions are space groups of the triclinic crystal system . Dih ( H ) is Abelian , with the semidirect product a direct product , iff all elements of H are their own inverse : Dih ( Z 1 ) = Dih 1 = Z 2 Dih ( Z 2 ) = Dih 2 = Z 2 × Z 2 ( Klein four-group ) Dih ( Dih 2 ) = Dih 2 × Z 2 = Z 2 × Z 2 × Z 2 etc . Topology Dih ( R n ) and its dihedral subgroups are disconnected topological groups . Dih ( R n ) consists of two connected components : the identity component isomorphic to R n , 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 . For the group Dih ∞ we can distinguish two cases : Dih ∞ as the isometry group of Z Dih ∞ as a 2D 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 . 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 Categories : Group theory | Finite groups | Euclidean symmetries In other languages : Deutsch | Français | Italiano | Polski 