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+---
+title: "Symmetric Groups in Python"
+date: 2019-05-22T20:02:21-04:00
+draft: false
+---
+
+**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](https://gist.github.com/Brandon-Rozek/adf9e1e64e2fbfcd3f8d3bc5da9322bf).
+
+#### 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.
+
+```python
+SymmetricElement(1,2,3)
+# array([[1., 2., 3.],
+#       [1., 2., 3.]])
+```
+
+```python
+SymmetricElement(1,2,3) * SymmetricElement(2,1,3)
+#array([[1., 2., 3.],
+#       [2., 1., 3.]])
+
+```
+
+```python
+SymmetricElement(1,2,3).inverse()
+#array([[1., 2., 3.],
+#       [1., 2., 3.]])
+```
+
+
+