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Dom::PermutationGroup -- permutation group

Introduction

Dom::PermutationGroup(n, generators <, options>) creates a domain for the group of permutations of {1,...,n} generated by generators.

Domain


Dom::PermutationGroup(n)

Parameters

n- positive integer
generators- list of permutations of {1,...,n}

Details

Introduction

The domain element Dom::PermutationGroup(n)(l) represents the bijective mapping of the first n positive integers that maps the integer i to l[i], for 1 ≤i ≤n.

Creating Elements


Dom::PermutationGroup(n)(l)

Parameters

l- list or array consisting of the first n integers in some order.

Categories

Cat::Group

Related Domains

Dom::DihedralGroup, Dom::SymmetricGroup, combinat::permutations

Entries

one

the identical mapping of the set {1, ..., n } to itself.

Method _mult: product of permutations

Method _invert: inverse of a permutation

Method func_call: function value of a permutation at a point

Method cycles: cycle representation of a permutation

Method order: order of a permutation

Method inversions: number of inversions

Method sign: sign of a permutation

Method random: random permutation

Method allElements: elements of the group

Method size: size of the group

Method convert: conversion of an object into a permutation

Method convert_to: conversion of a permutation into another type

Method expr: convert a permutation into a list

Example 1

We construct the cyclic group of order 4, generated by the cycle (1,2,3,4).:

>> G := Dom::PermutationGroup(4, [[[1,2,3,4]]])
     
                Dom::PermutationGroup(4, [[[1, 2, 3, 4]]])
        

It has 4 elements:

>> G::size;
   G::allElements();
     
                                     4
      
         [[2, 3, 4, 1], [3, 4, 1, 2], [4, 1, 2, 3], [1, 2, 3, 4]]
        

We construct a permutation of G by providing the images of 1, 2, etc.:

>> p := G([3,4,1,2]);
   p(1), p(2), p(3), p(4);
     
                               [3, 4, 1, 2]
      
                                3, 4, 1, 2
        

We construct a permutation from its cycle notation:

>> G([[1,2],[3,4]])
     
                               [2, 1, 4, 3]
        

Super-Domain

Dom::BaseDomain

Axioms

Ax::canonicalRep

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