# Documentation

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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`

## Axioms

`Ax::canonicalRep`

## 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)`