Dihedral group - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a dihedral group is the group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

of symmetries

Symmetry

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 perfection...

of a regular polygon

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

, including both rotations

Rotational symmetry

Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

and reflections

Reflection symmetry

Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

. Dihedral groups are among the simplest examples of finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

s, and they play an important role in group theory

Group theory

In mathematics and 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 groups endowed with additional operations and...

, geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, and chemistry

Chemistry

Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

.

See also: Dihedral symmetry in three dimensions

Dihedral symmetry in three dimensions

This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn .See also point groups in two dimensions.Chiral:...

Notation

There are two competing notations

Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...

for the dihedral group associated to a polygon with n sides. In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

the group is denoted D_n, while in algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

the same group is denoted by D_2n to indicate the number of elements.

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

Elements

A regular polygon with n sides has 2n different symmetries: n rotational symmetries

Rotational symmetry

and n reflection symmetries

Reflection symmetry

. The associated rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s and reflections

Reflection (mathematics)

In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...

make up the dihedral group D_n. If n is odd each axis of symmetry connects the mid-point of one side to the opposite vertex. If n is even there are n/2 axes of symmetry connecting the mid-points of opposite sides and n/2 axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether 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 D₈ on a stop sign

Stop sign

A Stop sign is a traffic sign to notify drivers that they must stop before proceeding.-Specifications:The Vienna Convention on Road Signs and Signals proposed standard stop sign diameters of 0.6, 0.9 or 1.2 metres. UK and New Zealand stop signs are 750, 900 or 1200 mm, according to sign...

:
The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections.

Group structure

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group

Group (mathematics)

.
The following Cayley table

Cayley table

A Cayley table, after the 19th century British mathematician Arthur Cayley, 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 multiplication table...

shows the effect of composition in the group D₃ (the symmetries of an equilateral triangle). 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.) Note that the composition operation is not commutative

Commutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

.

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

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

with modulus n.

Matrix representation

If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

. This lets us represent elements of D_n as matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, with composition being matrix multiplication

Matrix multiplication

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

.
This is an example of a (2-dimensional) group representation

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

.

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:

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.

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

Klein four-group

In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...

. Both are exceptional within the series:

They are abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

; for all other values of n the group Dih_n is not abelian.
They are not subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...

s of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

S_n, corresponding to the fact that 2n > 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

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...


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

Euclidean plane isometry

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...

which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions

Point groups in two dimensions

In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O, including O itself...

. D_n consists of n rotation

Rotation

s of multiples of 360°/n about the origin, and reflection

Reflection (mathematics)

s across n lines through the origin, making angles of multiples of 180°/n with each other. This is the symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

of a regular polygon

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

with n sides (for n ≥ 3; this extends to the cases n = 1 and n = 2 where we have a plane with respectively a point offset from the "center" of the "1-gon" and a "2-gon" or line segment).

Dihedral group D_n is generated

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

by a rotation r of order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

n and a reflection s of order 2 such that

In geometric terms: in the mirror a rotation looks like an inverse rotation.

In terms of complex numbers: multiplication by

and complex conjugation.

In matrix form, by setting

and defining

and

for

we can write the product rules for D_n as

(Compare coordinate rotations and reflections

Coordinate rotations and reflections

In geometry, 2D coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1...

.)

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 {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the y-axis.
D₂ is isomorphic

Group isomorphism

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. If there exists an isomorphism between two groups, then the groups are called isomorphic...

to the Klein four-group

Klein four-group

In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...

.

For n>2 the operations of rotation and reflection in general do not commute and D_n is not abelian

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

; 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

Symmetry

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 perfection...

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 2n elements of D_n can be written as e, r, r², ..., rⁿ⁻¹, s, r s, r² s, ..., rⁿ⁻¹ s. 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

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...

of O(2)

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

, 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

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

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

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

, octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

, and icosahedron

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

respectively).

Equivalent definitions

Further equivalent definitions of Dih_n are:

The automorphism group of the graph
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

consisting only of a cycle with n vertices (if n ≥ 3).
The group with presentation
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...

From the second presentation follows that Dih_n belongs to the class of Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

The semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...

of cyclic group
Cyclic group
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 g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

s Z_n and Z₂, with Z₂ acting on Z_n by inversion (thus, Dih_n always has a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

isomorphic to the group Z_n

is isomorphic to Dih_n if

is the identity and

is inversion.

Properties

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

Subgroup

of the symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

S_n via this permutation representation.

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

Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...

by −1, it is clear that it commutes with any linear transformation).

For odd n, abstract group Dih_2n is isomorphic with the direct product

Direct product of groups

In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

of Dih_n and Z₂.

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

If m divides

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

n, then Dih_n has n / m subgroup

Subgroup

s 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 divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

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.

Conjugacy classes of reflections

All the reflections are conjugate

Conjugacy class

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...

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. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon half the axes pass through two vertices, and half pass through two sides.

Algebraically, this is an instance of the conjugate Sylow theorem

Sylow theorem

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains...

(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 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 Dih_n is isomorphic to the affine group

Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....

Aff(Z/nZ)

and has order

where

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

, for k coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

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 2n, and for n even 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 (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

Dih₉ has 18 inner automorphism

Inner automorphism

In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...

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 automorphisms, e.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

Euler's totient function

In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n 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 n that...

, the multiplicative group of integers modulo n

Multiplicative group of integers modulo n

In modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it...

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).

Generalizations

There are several important generalizations of the dihedral groups:

The infinite dihedral group
Infinite dihedral group
In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups.-Definition:...

is an infinite group
Infinite group
In group theory, an area of mathematics, an infinite group is a group, of which the underlying set contains an infinite number of elements....

with algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s.
The orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

O(2), i.e. the symmetry group of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

, also has similar properties to the dihedral groups.
The family of generalized dihedral group
Generalized dihedral group
In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group O.-Definition:...

s includes both of the examples above, as well as many other groups.
The quasidihedral group
Quasidihedral group
In mathematics, the quasi-dihedral groups and semi-dihedral groups are non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of nonabelian groups of order 2n which have a cyclic subgroup of index 2...

s are family of finite groups with similar properties to the dihedral groups.

External links

Dihedral Group n of Order 2n by Shawn Dudzik, Wolfram Demonstrations Project
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...

.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.