Symmetric Group

## Description

The Sage command to create $S_5$ is `G = SymmetricGroup(5)`. Sage can represent elements of the symmetric group $S_n$ with either permutation notation or cycle notation. The element

\begin{align} \sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 1 & 5 & 4 \end{pmatrix} \end{align}

in $S_5$ can also be represented as $\sigma = (123)(45)$ using cycle notation. The Sage commands for representing $\sigma$ in permutation notation and cycle notation are `sigma = G([2,3,1,5,4])` and `sigma = G("(1,2,3)(4,5)")`, respectively.

In Sage, permutations are multiplied left to right.

## Sage Cell

## Options

```
G = SymmetricGroup(5)
G
```

```
sigma = G([2,3,1,5,4])
sigma
```

```
tau = G("(1,2,3)(4,5)")
tau
```

`sigma == tau`

```
rho = G("(1,2,3,5)")
rho
```

`rho * sigma`

`sigma * rho`

## Tags

Primary Tags: abstract algebra

Secondary Tags: permutations, symmetric group, permutation groups

## Related Cells

none

## Attribute

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Author: T. Judson

Date: 20 Jul 2017 13:54

Submitted by: Tom Judson