part3.md

title	date	author	mathjax	tags
Groups????	2014-07-13	colah	true	group theory, probability, convolution, math

$\newcommand{sq}[1]{ \raise{-1pt}{ \style{height: 15px; content: url('img/sqF-#1.png')}{ {\over~}\over~ } }~ } \newcommand{tri}[1]{ \raise{-1pt}{ \style{height: 15px; content: url('img/tri-#1.png')}{ {\over~}\over~ } }~ } \newcommand{Sq}[1]{ \raise{-3pt}{ \style{height: 30px; content: url('img/sq-#1.png')}{ \begin{array}{ccc} &&\~\\end{array} } } }!!$

Symmetry

Consider a square. Is it symmetric? How is it symmetric? How much symmetry does it have? What kind of symmetry does it have?

What do those questions even mean?

If you ask someone, they might tell you that a square has rotational symmetry. If you rotate a square by 90°, it's the same shape. Without knowing which corner was which, it would seem the exact same as it was before. You could lift it up, rotate it, and set it back down so that it covers the exact same space.

Let's call this rotation transformation $r$. To be precise, $r$ rotates a square clockwise by 90°. For example, $r\sq{e} = \sq{r}$. (The "F" on the square is there to let us determine orientation and see transformations.)

You might also be told that a square has horizontal symmetry or vertical symmetry. You can flip a square horizontally or vertically and still have a square. Let's focus on horizontal symmetry for now. We'll call horizontal flips $s$. $s$ performs a reflection across a vertical line through the middle of the square. For example, $s\sq{e} = \sq{s}$.

We now have two transformations, $r$ and $s$, which transform squares into another square of the same shape. It turns out that these two transformations form a kind of "basis" for all the others. By using them in some pattern, you can build the other transformations, like vertical flipping.

Starting with our original square $\sq{e}$ in the bottom left corner, the following graph shows the transformed versions generated by combining $r$ and $s$ in different ways. $r$ and $s$ are represented by arrows of different colors. $r$ arrows are colored blue and $s$ arrows are colored red.

We can use the graph to investigate what happens if we perform a sequence of transformations. For example, what happens if we rotate, flip and then rotate again? Well, we start at our original square, $\sq{e}$, and trace: $\sq{e} \xrightarrow{r} \sq{r} \xrightarrow{s} \sq{r3s} \xrightarrow{r} \sq{s}$. In the end, we're left with just horizontally flipped version of the original, $s\sq{e} = \sq{s}$. If we want to express this surprising fact, we can use multiplication like notation: $rsr \sq{e} = s \sq{e}$.

If we want to think about our graph a bit more abstractly, we can factor apart the transformation and the original square (eg. $\sq{r2s} \to r^2s\sq{e}$).

Here, $e$ is the identity element: for all $X$, $eX = X$. For example $e\sq{e} = \sq{e}$.

We can go a bit further. The original square, $\sq{e}$, seems a bit unnecessary in $rsr \sq{e} = s \sq{e}$. Why not just say $rsr = s$? We can just drop the factored out $\sq{e}$, both in equations and our graph.

Now, here's the essential realization: $r$ and $s$ could have been other things and we would have had the exact same graph. $r$ could have been rotating 90° counterclockwise. $s$ could have been vertical flips. Or we could have been transforming an entirely different kind of object. All that matters is the relationship between $r$ and $s$, how they interact. What we saw with the squares was just one particular way this graph, this abstract pattern, could appear in the real world.

Mathematicians call these abstract patterns groups. There is an entire field of math dedicated to them. Connections between a group and an object like the square are called group actions.

But... What is a group?

Not all graphs are groups. Only a very special kind of graph is.

Firstly, the graph is directed (the edges are arrows) and has colored edges. At every vertex, exactly one arrow of a given color comes out and one goes in.

But the key property of these graphs is more subtle. We created our graph by starting with an original square, $\sq{e}$. But what if we said the original square was $\sq{s} = s\sq{e}$?

Which position we say is the "initial" position is arbitrary. No matter which position you think of as the initial one, the graph is the same. The graph is perfectly symmetrical, in some sense.¹

One important idea is that, given two elements of the group (as we call the vertices of the graph), we can combine them together. Sometimes we make analogies to addition and write combining two elements $a$ and $b$ as $a+b$, while other times we make analogies to multiplication and write $a\cdot b$.

"Adding" or "multiplying" two group elements is very similar to vector addition. We decide that one point on the graph is our identity element (the original position), and find the two elements we want to multiply, $a$ and $b$. We pick paths from the identity to $a$ and $b$. Then we stick the $a$ path on to the end of $b$, to bring us to $a+b$ or $a\cdot b$ (depending on the chosen notation).

The Algebraic Perspective

FIX diagram

The above is almost unrecognizable as group theory, from a traditional perspective.

The traditional perspective on group theory is extremely different than the above. Historically, group theory was motivated by a desire to abstract

Group theory arose from what computer scientists would now call polymorphism

...

We could define a group to have more or less requirements. If it was weaker, had less requirements, more kinds of objects would be groups and the results we prove about groups would be more broadly applicable. If it was stronger, had more requirements, we would be talking about a more specific kind of object and could prove more about them. In mathematics one often balances generality and specificity like this.

Mathematicians study both weaker and stronger versions of groups. But, somehow, groups are special. They aren't too hot, they aren't too cold: they're just right.

Footnotes

1. Note that the graph embedding isn't necessarily symmetrical. ↩