## `Dom::DihedralGroup` -- dihedral groups

#### Introduction

`Dom::DihedralGroup``(n)` creates the group of all congruent mappings of the plane that induce a bijective mapping of the set of corners of a regular n-angle to itself.

#### Domain

`Dom::DihedralGroup(n)`

#### Parameters

`n` | - | positive integer

#### Introduction

`Dom::DihedralGroup``(n)([a,b])` represents the group element ``ta carried out after rb'', where r is a rotation that maps each corner to its left neighbor, and t is a reflection w.r.t. some fixed central diagonal.

#### Creating Elements

``Dom::DihedralGroup`n(l)`

#### Parameters

`l` | - | list or array of two integers

#### Categories

`Cat::Group`

#### Related Domains

`Dom::PermutationGroup`

#### Entries

size

  the number of elements, which equals 2n.

one

  the mapping leaving each point fixed.

#### Method _mult: functional composition of elements

- `_mult(dom a...)`
- - The product of elements of `Dom::DihedralGroup` is defined as their functional composition, with the factors applied from right to left.
  - This method overloads the kernel function `_mult`.

#### Method _invert: inverse of an element

- `_invert(dom a)`
- - The inverse of `a` is defined to be the mapping that sends every corner to its pre-image under `a`. (This agrees with the usual notion of the inverse of a bijective mapping.)
  - This method overloads the kernel function `_invert`.

#### Method _power: power of an element

- `_power(dom a, integer n)`
- - This method computes an.
  - It overloads the kernel function `_power`.

#### Method order: order of a group element

- `order(dom a)`
- - This method returns the smallest positive integer k for which ak=1.

#### Method random: random element

- `random()`
- - This method returns a random element of the group; the results are uniformly distributed.

#### Method expr: convert group element to list

- `expr(dom a)`
- - This method returns a list of two elements; the meaning is the same as in the case of creating elements of `Dom::DihedralGroup`.

#### Method TeX: TeX output of a group element

- `TeX(dom a)`
- - The element `a` is returned as a TeXstring generated from its list representation. This avoids using fixed names for the generators, as there is no standard for them in the literature.

#### Example 1

Define the group D_6, i.e., the group of congruence mappings of the hexagon:

    >> G := Dom::DihedralGroup(6)

                               Dom::DihedralGroup(6)

Then elements may be created as follows:

    >> a := G([7, 19]);

                                      [1, 1]

This means that 19 rotations--mapping each corner to its left neighbor--and 7 reflections have the same effect as one operation of either type.

#### Super-Domain

`Dom::BaseDomain`

#### Axioms

`Ax::canonicalRep`

#### Changes

- no changes

Do you have questions or comments?
Copyright © SciFace Software GmbH & Co. KG 2000