# Symmetric Groups in Python

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3 minute reading time

**Warning:** This post is meant for someone whose familiar with concepts of Abstract Algebra.

## Refresher

### Definitions

An **operation** on a set is a calculation that maps one element in a set onto another element of the set.

A **group** in mathematics is a set and an operation that follows the three properties:

- There exists an identity element.
- The operation is associative.
- For every element, there exists an inverse of that element in the set.

**Symmetric Groups** are groups whose elements are all bijections from the set onto itself and operation which is composition of functions.

### Example

Let’s look at the group $\mathbb{Z}_3$. Here is an example of an element of its symmetric group. $$ \begin{pmatrix} 0 & 1 & 2 \ 1 & 2 & 0 \end{pmatrix} $$ This element maps $0 \rightarrow 1$, $1 \rightarrow 2$, and $2 \rightarrow 0$.

A good way to check if something similar to the above is an element of a symmetric group is pay attention to the second row. Make sure that it only contains the elements of the set you care about (ex: $\mathbb{Z}_3$) and that there are no repeats.

Let’s look at an example of composing two elements from this symmetric group.

$$ \begin{pmatrix} 0 & 1 & 2 \ 1 & 2 & 0 \end{pmatrix} \circ \begin{pmatrix} 0 & 1 & 2 \ 0 & 2 & 1 \ \end{pmatrix} = \begin{pmatrix} 0 & 1 & 2 \ 1 & 0 & 2 \ \end{pmatrix} $$

The main thing to remember here is that you must compose from right to left.

$0 \rightarrow 0$ and then $0 \rightarrow 1$, so ultimately $0 \rightarrow 1$.

$1 \rightarrow 2$ and $2 \rightarrow 0$, so ultimately $1 \rightarrow 0$.

$2 \rightarrow 1$ and $1 \rightarrow 2$, so ultimately $2 \rightarrow 2$.

### Finding Inverses

Finding the inverse is simple, since all you need to do is flip the two rows and sort it again. $$ \begin{pmatrix} 0 & 1 & 2 \ 1 & 2 & 0 \end{pmatrix}^{-1} = \begin{pmatrix} 1 & 2 & 0 \ 0 & 1 & 2 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 2 \ 2 & 0 & 1 \end{pmatrix} $$

### Code Implementation

For Abstract Algebra homework, there was a lot of compositions of these symmetric elements. Sadly, I get pretty lazy doing these by hand for many hours. So like any Computer Scientist, I created a simple script in Python to help me compute these.

The code is located in this gist.

#### Basic Usage

`SymmetricElement`

takes in the second row of the matrices we were playing with. You can find the inverse with `element.inverse()`

and you can compose two symmetric elements together with the `*`

operation.

```
SymmetricElement(1,2,3)
# array([[1., 2., 3.],
# [1., 2., 3.]])
```

```
SymmetricElement(1,2,3) * SymmetricElement(2,1,3)
#array([[1., 2., 3.],
# [2., 1., 3.]])
```

```
SymmetricElement(1,2,3).inverse()
#array([[1., 2., 3.],
# [1., 2., 3.]])
```