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110 lines
2.9 KiB
Markdown
110 lines
2.9 KiB
Markdown
---
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title: "Symmetric Groups in Python"
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date: 2019-05-22T20:02:21-04:00
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draft: false
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tags: [ "abstract algebra" ]
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---
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**Warning:** This post is meant for someone whose familiar with concepts of Abstract Algebra.
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## Refresher
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### Definitions
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An **operation** on a set is a calculation that maps one element in a set onto another element of the set.
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A **group** in mathematics is a set and an operation that follows the three properties:
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- There exists an identity element.
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- The operation is associative.
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- For every element, there exists an inverse of that element in the set.
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**Symmetric Groups** are groups whose elements are all bijections from the set onto itself and operation which is composition of functions.
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### Example
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Let's look at the group $\mathbb{Z}_3$. Here is an example of an element of its symmetric group.
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$$
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\begin{pmatrix}
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0 & 1 & 2 \\
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1 & 2 & 0
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\end{pmatrix}
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$$
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This element maps $0 \rightarrow 1$, $1 \rightarrow 2$, and $2 \rightarrow 0$.
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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.
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Let's look at an example of composing two elements from this symmetric group.
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$$
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\begin{pmatrix}
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0 & 1 & 2 \\
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1 & 2 & 0
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\end{pmatrix}
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\circ
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\begin{pmatrix}
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0 & 1 & 2 \\
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0 & 2 & 1 \\
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\end{pmatrix}
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=
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\begin{pmatrix}
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0 & 1 & 2 \\
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1 & 0 & 2 \\
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\end{pmatrix}
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$$
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The main thing to remember here is that you must compose from right to left.
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$0 \rightarrow 0$ and then $0 \rightarrow 1$, so ultimately $0 \rightarrow 1$.
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$1 \rightarrow 2$ and $2 \rightarrow 0$, so ultimately $1 \rightarrow 0$.
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$2 \rightarrow 1$ and $1 \rightarrow 2$, so ultimately $2 \rightarrow 2$.
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### Finding Inverses
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Finding the inverse is simple, since all you need to do is flip the two rows and sort it again.
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$$
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\begin{pmatrix}
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0 & 1 & 2 \\
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1 & 2 & 0
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\end{pmatrix}^{-1} =
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\begin{pmatrix}
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1 & 2 & 0 \\
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0 & 1 & 2
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\end{pmatrix} =
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\begin{pmatrix}
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0 & 1 & 2 \\
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2 & 0 & 1
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\end{pmatrix}
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$$
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### Code Implementation
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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.
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The code is located in [this gist](https://gist.github.com/Brandon-Rozek/adf9e1e64e2fbfcd3f8d3bc5da9322bf).
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#### Basic Usage
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`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.
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```python
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SymmetricElement(1,2,3)
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# array([[1., 2., 3.],
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# [1., 2., 3.]])
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```
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```python
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SymmetricElement(1,2,3) * SymmetricElement(2,1,3)
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#array([[1., 2., 3.],
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# [2., 1., 3.]])
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```
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```python
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SymmetricElement(1,2,3).inverse()
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#array([[1., 2., 3.],
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# [1., 2., 3.]])
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```
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