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

#### Code

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


None

## Tags

Primary Tags: abstract algebra

Secondary Tags: permutations, symmetric group, permutation groups

none