Numbers measure size, groups measure symmetry – M.A. Armstrong: Groups and Symmetry

Group theory is the mathematical tool one uses in order to work with symmetries. Because symmetries are defined as invariance under transformations, one defines a group as a collection of transformations. Let’s get started with two easy examples to get a feel for what we want to do:

1) A square is mathematically a set of points (for example, the four corner points are part of this set) and a symmetry of the square is a transformation that maps this set of points into itself.

Examples of symmetries of the square are certain (not all!) rotations about the origin. Certain rotations about the origin map the  square into itself. This means, they map every point of the set to a point that lies again in the set, and the one says, the set is invariant under such a transformation.

This becomes obvious if we focus on the corner points of the square. Transforming the set by a clockwise rotation by, say 5°, maps these points into points outside the original set that defines
the square. For instance, the corner point $A$ is mapped to the point $A’$, which is not found inside the set, that defined the square in the first place. Therefore a rotation by 5° is not a symmetry of the square. Of course, the rotated object is still a square, but not the same square (read: set of points). Nevertheless, a clockwise rotation by 90° is a symmetry of the square because the point $A$ is mapped to the point $B$, which lies again in the original set. Other examples of symmetry transformations of the square would be rotations by 180°, 270° and of course 0°.

Another perspective: Imagine you close your eyes for a moment, and then someone transforms the square in front of you. If you can’t tell after opening your eyes again if the other person changed anything at all, the transformation the person performed was a symmetry transformation.

The set of transformations that leave the square invariant is called a group. The transformation parameter, here the rotation angle, can’t take on arbitrary values and the group is called a discrete group.

2) Another example is the set of transformations that leave the unit circle invariant. Again, the unit circle is defined as a set of points and a symmetry transformation is a map, that maps this set into itself. 

The unit circle is invariant under all rotations about the origin, not just a few. In other words: the transformation parameter (the rotation angle) can take on arbitrary values, and the group is said to be a continuous group.

Of course, mathematics isn’t exclusively about geometric shapes and one can find symmetries of different kinds of objects, too. For instance, considering vectors, one can look at the set of transformations that leave the length of any vector unchanged. (The object that is unchanged in this case is the metric, which is the mathematical object that defines length.) For this reason, the definition of symmetry I gave at the beginning was very general: Symmetry means invariance under a transformation. Luckily, there is one mathematical tool, called group theory, that lets us work with all kinds of symmetries. (As a side note: group theory was invented historically to investigate symmetries of equations).

To make the idea of a mathematical tool that lets us deal with symmetries precise, we need to distill the defining features of symmetries in a mathematical form:

• • Leaving the object in question unchanged (“doing nothing”) is always a symmetry and therefore, every group needs to contain an identity element. In the examples above, the identity element is the rotation by 0°.

• • Transforming some object and performing afterwards the inverse transformation must be equivalent to doing nothing. Therefore, there must be, to every element in the set, an inverse element. A transformation followed by its inverse transformation is, by definition of the inverse transformation, the same as the identity transformation. In the above examplesthis means that the inverse transformation to a rotation by 90° is a rotation by -90°. A rotation by 90° followed by a rotation by -90° is the same as a rotation by 0°.

• • Performing a symmetry transformation followed by a second symmetry transformation is again a symmetry transformation. A rotation by 90° followed by a rotation by 180° is a rotation by 270°, which is a symmetry transformation, too. This property is called closure.

• • The combination of transformations must be associative. A rotation by 90° followed by a rotation by 40°, followed by a rotation by 110° is the same as a rotation by 130° followed by a rotation by 110°, which is the same as a rotation by 90° followed by a rotation by 150°. In a symbolic form: $R(110°) R(40°) R(90°)=  R(110°) \big(R(40°) R(90°)\big)= R(110°) R(130°)$  and $R(110°) R(40°) R(90°)=\big(R(110°) R(40°) \big) R(90°)= R(150°) R(90°)=$ This is called associativity. (This may not be confues with commutativity, because a the elements of a group, in general, do not commute. For example rotations around different axes: $R_x(30^\circ) R_z(40^\circ) \neq R_z(40^\circ) R_x(30^\circ) $)

• • To be able to talk about the things above one needs a rule, to be precise: a binary operation, for the combination of group elements. In the above examples, the standard approach would be to use rotation matrices and the rule for combining the group elements (the corresponding rotation matrices) would be ordinary matrix multiplication. Nevertheless there are often different ways of describing the same thing. The rotations in the plane can be described by multiplication with unit complex numbers, too. Therefore, the rule for combining group elements would be complex number multiplication. Happily, group theory lets one study such, maybe confusing, diversity in a very systematic way. This branch of group theory is called representation theory, and we will have a look at it in another post.

We are now able to see that the abstract definition of a group simply states (obvious) properties of symmetry transformations:

A group is a set $G$, together with a binary operation $\circ$ defined on $G$ that satisfies the following axioms

•  Closure: For all $g_1, g_2 \in G$, $g_1 \circ g_2 \in G$
• Identity: There exists an identity element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ G$
• Inverses: For each $g \in G$, there exists an inverse element $g^{-1} \in G$ such that $g \circ g^{-1}=e = g^{-1} \circ $ g.
• Associativity: For all $g_1, g_2, g_3 \in G$, $g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3$.

Group theory is the branch of mathematics one uses to work with symmetries.  A symmetry of an object is a transformation that leaves the object unchanged. The word object is chosen purposefully, because it is very vague. There is one branch of mathematics that deals with all kinds of symmetries, any kind of object can have.

The most familiar type of symmetries that come to one’s mind are symmetries of geometric shapes, so lets start with that.

Symmetries of Geometric Shapes

A square is defined mathematically as a set of points. A symmetry of the square is a transformation that maps this set of points into itself. This means concretely that by the transformation, no point is mapped to a point outside of the set that defines the square.

Obvious examples of such transformations are rotations, by $90^\circ$, $180^\circ$, $270^\circ$, and of course $0^\circ$.

A counter-example is a rotation by, say $5^\circ$. The upper-right corner point $A$ of the square is obviously mapped to a point $A’$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^\circ$ is no symmetry of the square.

A characteristic property of the symmetries of the square is that the combination of two transformations that leave the square invariant, is again a symmetry. For example, combining a rotation by $90^\circ$ and $180^\circ$ is equivalent to a rotation by $270^\circ$, which is again a symmetry of the square. We will elaborate on this in the next post. In fact, the basic axioms of group theory can be derived from such an easy example.

Next, lets turn to something completely different.

Symmetries of Numbers?

Take a look at the fourth roots of unity, i.e., the (complex) numbers $z$ that give $1$ when raised to the fourth power:

\begin{equation} z^4 \stackrel{!}{=}1. \end{equation}

We can compute the solutions using the De Moivre Formula:

\begin{equation} z^n= (\cos(x)+i\sin(x))^n= \cos(nx)+i\sin(nx) \end{equation}

which follows when we write the complex number, using the Euler’s formula, as $\mathrm{e}^{ix}=\cos(x)+i\sin(x)$.  The De Moivre Formula here yields,

\begin{equation} z^4= \cos(4x)+i\sin(4x)  \stackrel{!}{=}1. \end{equation}

We know $\cos(2πk) + i \sin(2πk)=1$ for any integer k. The fourth roots of unity are

\begin{equation} z_k= \cos(2πk/4) + i \sin(2πk/4) \qquad \mathrm{ \   for}  \qquad  k = 0,1,2,3. \end{equation}

\begin{equation}  \rightarrow z_0 = 1 \qquad , \qquad  z_1 = i  \qquad , \qquad z_2= -1 \qquad , \qquad z_3 = -i \qquad , \qquad \end{equation}

We can find a curious property of these numbers: The multiplication of two such solutions result again in a solution. Drawing the solutions in the complex plane gives us a geometric interpretation of this fact. The multiplication rotates any complex number by exactly the correct amount into another solution. The set of solutions is said to be closed under multiplication: By mere multiplication we can’t get any number that doesn’t lies in this set.

We have thus found a very similar structure in two completely different objects. On the one hand symmetries of the square, on the other hand the fourth roots of unity. Both sets contain exactly 4 objects and both share the property called closure: the combination of two objects in this set, lies again in the set.  This indicates that there might be a map between the two sets, and indeed rotations in two-dimensions can be described by complex number multiplication. Exploring such intersections and structures in a systematic way is what group theory is all about.

Closure holds in general, for the n-th roots of unity, because, given two n-th roots $z,z’$, i.e. numbers for which $z^n=1$ and $z’^n=1$ holds, we have

\begin{equation}  (z z’)^n = z^n  z’^n = 1 \cdot 1 = 1 \quad \end{equation}

It’s no surprise that there exists a similar close relationship between the fifth-roots of unity and the symmetries of the pentagon.

We can search for higher order roots of unity and draw them in the complex plane.  For example, the 100-th roots of units, i.e., the complex numbers satisfying $z^100 =1$, look like this:

Copyright by Wolfram|Alpha

These numbers are again closely related to the symmetries of a geometric object, that I don’t know the name of. Nevertheless, we can see where we are heading if we choose $n$ in $z^n=1$ to be larger and larger. Finally, we arrive at the unit circle,

The circle is quite different from the square and the pentagon, because the circle has infinitely many symmetries. Any rotation about the origin maps the circle into the circle. Such symmetries, called continuous symmetries, are the topic of a special branch of group theory, called Lie theory (after its inventor Sophus Lie).

Mathematics is a vast field. What started with integers and geometric shapes 2000 years ago, has become incredible diversified. Particular interesting things often happen at the intersection of branches of mathematics that don’t seem to have anything in common. Group theory is a framework that helps exploring such intersections.