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

(1)
\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

Permalink:

Author: T. Judson

Date: 20 Jul 2017 13:54

Submitted by: Tom Judson

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License