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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::DihedralGroupn(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

Method _invert: inverse of an element

Method _power: power of an element

Method order: order of a group element

Method random: random element

Method expr: convert group element to list

Method TeX: TeX output of a group element

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




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