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Dom::SymmetricGroup

Symmetric groups

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Syntax

Dom::SymmetricGroup(n)
Dom::SymmetricGroup(n)(l)

Description

Dom::SymmetricGroup(n) creates the symmetric group of order n, that is, the domain of all the permutations of {1, …, n} elements.

A permutation of n elements is a bijective mapping of the set {1, …, n} onto itself.

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

Superdomain

Dom::BaseDomain

Categories

Cat::Group

Examples

Example 1

Consider the group of permutations of the first seven positive integers:

G := Dom::SymmetricGroup(7)

We create an element of G by providing the image of 1, 2, etc.:

a:=G([2,4,6,1,3,5,7])

a(3)

Parameters

n

Positive integer

l

List or array consisting of the first n integers in some order.

Entries

"one"

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

Methods

expand all

Mathematical Methods

_mult(a1, …)

This method overloads the function _mult.

_invert(a)

This method overloads the function _invert.

func_call(a, i)

It computes the function value of a at i, i.e., the integer that i is mapped to by the permutation a; i must be an integer between 1 and n.

cycles(a)

order(a)

inversions(a)

sign(a)

random()

Access Methods

allElements()

size()

Conversion Methods

convert(x)

convert_to(a, T)

expr(a)

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