Group Theory Overview

Group Theory: The Mathematical Language of Symmetry

Notes

A group is an algebraic object that measures symmetry.
There is an abstract and a "dynamic" perspective on groups. We need both, and it is useful to play the two off of each other.
A group is rigorously defined as a set $G$ $G$ together with a binary multiplication $G\times G \to G$ $G \times G \to G$ satisfying the axioms:
- associativity: $(ab)c = a(bc)$
- identity: there's some element $e$ such that $ex = x = xe$ for all $x\in G$
- invertibility: for every $g\in G,$ there's some $h\in G$ such that $gh = e = hg$
We will see more of this later.
Groups were originally invented to understand symmetries of polynomials. For example, in the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ we can take either plus or minus the square root, and neither one is "better" than the other. The original application of this was to show that the quintic equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$ does not have a general solution by radicals.
Groups are ubiquitous throughout mathematics.
There is a dichotomy between the study of finite groups and the study of infinite groups. In the latter case, one must usually use geometric tools to understand the groups in question.

Problems

Finish filling in the multiplication table for the symmetry group of the triangle (this group is called $\Dihed 3$). You can use the widget to help you.

$\circ$	$r$	$f_1$
$e$
$r$		$f_3$
$R$
$f_1$	$f_2$
$f_2$		$R$
$f_3$

Answer Solution

$\circ$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$e$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$r$	$r$	$R$	$e$	$f_3$	$f_1$	$f_2$
$R$	$R$	$e$	$r$	$f_2$	$f_3$	$f_1$
$f_1$	$f_1$	$f_2$	$f_3$	$e$	$r$	$R$
$f_2$	$f_2$	$f_3$	$f_1$	$R$	$e$	$r$
$f_3$	$f_3$	$f_1$	$f_2$	$r$	$R$	$e$

Using the widget to help us, we can eventually arrive at the following table by brute force:

$\circ$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$e$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$r$	$r$	$R$	$e$	$f_3$	$f_1$	$f_2$
$R$	$R$	$e$	$r$	$f_2$	$f_3$	$f_1$
$f_1$	$f_1$	$f_2$	$f_3$	$e$	$r$	$R$
$f_2$	$f_2$	$f_3$	$f_1$	$R$	$e$	$r$
$f_3$	$f_3$	$f_1$	$f_2$	$r$	$R$	$e$

However, thinking about the specifics of the problem at hand reveals several shortcuts we can take.

Associativity

Suppose that our current knowledge of the multiplication table is

$\circ$	$r$	$f_1$	$f_3$
$e$
$r$		$f_3$
$R$
$f_1$	$f_2$		$?$
$f_2$		$R$
$f_3$

and we wish to determine $f_1\circ f_3.$ Since we know that $f_3 = r\circ f_1,$ we can rewrite this as $f_1\circ r\circ f_1.$ But we already know that $f_1\circ r = f_2,$ so this in turn becomes $f_2\circ f_1,$ which we know to be $R.$ Altogether, and being more careful about substitution, this looks like $f_1\circ f_3 = f_1\circ(r\circ f_1) = (f_1 \circ r)\circ f_1 = f_2\circ f_1$ The middle step—where we moved the parentheses around—is exactly the associativity property of groups; it means that it is safe to write the expression $f_1\circ r\circ f_1$ without parentheses, since either way of parenthesizing yields the same result.

Invertibility and "process of elimination"

We can also save time using process of elimination. To wit, suppose we figure out most of the $R$ row of the table:

$\circ$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$R$	$R$	$e$	$r$	$f_2$	$f_3$	$?$

We might conclude "by process of elimination" that the remaining entry has to be $f_1,$ since (a) all other elements have already been used in this row, while (b) $f_1$ has not yet appeared. This is a familiar strategy in puzzle games like Sudoku—but why is it legitimate here?

First, I claim that the following (hypothetical) statements are equivalent (meaning the truth of any one of them implies the truth of the others):

Every group element appears at least once in every row/column.
Every group element appears no more than once in every row/column (that is, there are no repetitions).
Every group element appears exactly once in every row/column.

(Think about this: it is because there are the same number of spaces to fill as there are things we can put in them. This is a fundamental fact about finite sets which will reappear several times in different guises.) When we use "process of elimination", we are relying on the first of these to be true. We will show the second of these to be true; since the three statements are equivalent, this will justify the use of elimination.

Showing that no group element appears more than once in row $x$ means showing that if $x\circ y_1 = x\circ y_2,$ then we must have $y_1 = y_2.$ Since we are working in a group, there is some element $x^{-1}$ that "undoes" $x,$ in the sense that $x^{-1}\circ x = e.$ Multiplying our equation by $x^{-1},$ we get $\begin{aligned} x\circ y_1 &= x\circ y_2\\ x^{-1}\circ (x\circ y_1) &= x^{-1}\circ (x\circ y_2)\\ (x^{-1}\circ x)\circ y_1 &= (x^{-1}\circ x)\circ y_2\\ e\circ y_1 &= e\circ y_2\\ y_1 &= y_2 \end{aligned}$ Here we have used the group property that every element has an inverse.

Orientation

The previous two strategies work for any group. There is a more sophisticated strategy which is specific to the particular kind of group we are analyzing. Suppose we have figured out this much of the $f_2$ row:

$\circ$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$f_2$	$f_2$	$f_3$	$?$	$?$	$?$	$?$

and we want to know $f_2\circ R.$ At first blush, we cannot use elimination to solve this: it could be any of $e, r, R, f_1.$ However, each state of the triangle has one of two possible orientations. If the triangle were painted different colors on either side, one could identify the orientation with which color was visible; in our case, we can see it according to whether the numbers 1, 2, 3 go counterclockwise or clockwise around the circle.

In turn, the symmetries of the triangle can be classed as either orientation-preserving or orientation-reversing: $e, r, R$ preserve the orientation, while $f_1,f_2,f_3$ change it. If we color the cells of the table green for orientation-preserving and red for orientation-reversing, we obtain the following.

$\circ$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$e$
$r$
$R$
$f_1$
$f_2$
$f_3$

In particular, our knowledge of the $f_2$ row now looks like

$\circ$	$e$	$r$	$R$	$f_1$	$f_2$	$f_3$
$f_2$	$f_2$	$f_3$	$?$	$?$	$?$	$?$

We see that $f_2\circ R$ must be orientation-reversing, which rules out $e,$ $r,$ and $R.$ Thus we must have $f_2\circ R = f_1.$

The concept of "orientation" is a very important one that we will encounter frequently, particularly in calculus, geometry, and topology. New perspectives on the idea of "orientation" remain the subject of current research and memes.

Vistas

Classification of finite simple groups

For a natural number $n,$ there is a finite (although possibly very large) number of groups of order $n.$ One can try to classify all the finite groups: this means, for each $n,$ giving a list of all the groups of order $n,$ or at least having a fairly explicit description of them. One can make some progress towards this goal: if $p$ is a prime number then the only group of order $p$ is the "rotational" group $\Cycl p,$ while the two different groups of order 6 we encountered are the only ones possible.

In general, however, this problem is not tractable: there are just way too many possibilities to get a handle on. There is a useful analogy to be made with chemistry: there are too many possible molecules for us to easily list them all. However, molecules are made up of atoms, and we can classify all the atoms: this is the periodic table of the elements. The problem is then reduced to understanding, for any given fixed collection of atoms, the different ways they can combine to form a molecule.

In group theory, one can break complicated groups up into smaller groups; groups that can't be broken down any further are called simple groups. The name "simple" does not mean that they are easy to understand, just that they can't be broken down any further—they are usually very complicated. Thus simple groups are the "atoms", while general groups are the "molecules". The "atoms" making up a group are called its composition factors. In the video, we encountered two groups of order 6: the symmetry group $\Dihed 3$ of the triangle, and the symmetry group $\Cycl 6$ of the colored hexagon. These have the same composition factors, namely a $\Cycl 2$ and a $\Cycl 3,$ but these building blocks are put together in different ways. Thus one can separate the problem of classifying groups in to two parts.

Determine all the finite simple groups; this can be thought of as the "periodic table" of groups.
Given any set of finite simple groups, determine the ways they can combine to form a group.

Both of these problems are very hard. However, after much effort the first step was completed in 2004. This is a major achievement of twentieth-century mathematics. The finite simple groups can be organized into three infinite families, as well as 27 "sporadic" outliers. These sporadic groups have mysterious connections to number theory and physics.

Video Notes Exercises Vistas