Groups

Groups are some of the most basic concepts in mathematics. They are even more basic than the things we discussed in Rings, Fields, and Ideals. In fact, all these things require the concept of groups before they can even be defined rigorously. But apart from being a basic stepping stone toward other concepts, groups are also extremely useful on their own. They can be used to represent the permutations of a set. They can also be used to describe the symmetries of an object. Since symmetries are so important in physics, groups also play an important part in describing physical phenomena. The standard model of particle physics, for example, which describes the fundamental building blocks of our physical world such as quarks, electrons, and photons, is expressed as a “gauge theory” with symmetry group $U(1)\times SU(2)\times SU(3)$.

We will not discuss something of this magnitude for now, although perhaps in the future we will (at least electromagnetism, which is a gauge theory with symmetry group $U(1)$). Our intention in this post will be to define rigorously the abstract concept of groups, and to give a few simple examples. Whatever application we have in mind when we have the concept of groups, it will have the same rigorous definition, and perhaps express the same idea at its very core.

First we will define what a law of composition means. We have been using this concept implicitly in previous posts, in concepts such as addition, subtraction, and multiplication. The law of composition makes these concepts more formal. We quote from the book Algebra by Michael Artin:

A law of composition is a function of two variables, or a map

$\displaystyle S\times S\rightarrow S$

Here $S\times S\rightarrow S$ denotes, as always, the product set, whose elements are pairs $a, b$ of elements of $S$.

There are many ways to express a law of composition. The familiar ones include

$\displaystyle a+b=c$

$\displaystyle a\circ b=c$

$\displaystyle a\times b=c$

$\displaystyle ab=c$

From the same book we now quote the definition of a group:

A group is a set $G$ together with a law of composition that has the following properties:

• The law of composition is associative: $(ab)c=a(bc)$ for all $a$$b$, and $c$.
• $G$ contains an identity element $1$, such that $1a=a$ and $a1=a$ for all $a$ in $G$.
• Every element $a$ of $G$ has an inverse, an element $b$ such that $ab=1$ and $ba=1$.

Note that the definition has used one particular notation for the law of composition, but we can use different symbols for the sake of convenience or clarity. This is merely notation and the definition of a group does not change depending on the notation that we use.

All this is rather abstract. Perhaps things will be made clearer by considering a few examples. For our first example, we will consider the set of permutations of the set with three elements which we label $1$, $2$, and $3$. The first permutation is what we shall refer to as the identity permutation. This sends the element $1$ to $1$, the element $2$ to $2$, and the element $3$ to $3$.

Another permutation sends the element $1$ to $2$, the element $2$ to $1$, and the element $3$ to $3$. In other words, it exchanges the elements $1$ and $2$ while keeping the element $3$ fixed. There are two other permutations which are similar in a way, one which exchanges $2$ and $3$ while keeping $1$ fixed, and another permutation which exchanges $1$ and $3$ while keeping $2$ fixed. To more easily keep track of these three permutations, we shall refer to them as “reflections”.

We have now enumerated four permutations. There are two more. One permutation sends $1$ to $2$$2$ to $3$, and $3$ to $1$. The last permutation sends $1$ to $3$$2$ to $1$, and $3$ to $2$. Just as we have referred to the earlier three permutations as “reflections”, we shall now refer to these last two permutations as “rotations”.

We now have a total of six permutations, which agrees with the result one can find from combinatorics. Our claim is that these six permutations form a group, with the law of composition given by performing first one permutation followed by the other. Therefore the reflection that exchanges $2$ and $3$, followed by the reflection that exchanges $1$ and $3$, is the same as the rotation that sends $1$ to $3$$2$ to $1$, and $3$ to $2$, as one may check.

We can easily verify two of the properties required for a set to form a group. There exists an identity element in our set of permutations, namely the identity permutation. Permuting the three elements $1$, $2$, and $3$ via the identity permutation (i.e. doing nothing) followed by a rotation or reflection is the same as just applying the rotation or reflection alone. Similarly, applying a rotation or reflection, and then applying the identity permutation is the same as applying just the rotation or reflection alone.

Next we show that every element has an inverse. The rotation that sends $1$ to $2$$2$ to $3$, and $3$ to $1$ followed by the rotation that sends $1$ to $3$$2$ to $1$, and $3$ to $2$ results in the identity permutation. Also the rotation that sends $1$ to $3$$2$ to $1$, and $3$ to $2$ followed by the rotation that sends $1$ to $2$$2$ to $3$, and $3$ to $1$ results in the identity permutation once again. Therefore we see that the two rotations are inverses of each other. As for the reflections, we can see that doing the same reflection twice results in the identity permutation. Every reflection has itself as its inverse, and of course the same thing holds for the identity permutation.

The associative property holds for the set of permutations of three elements, but we will not prove this statement explicitly in this post, as it is perhaps best done by figuring out the law of composition for all the permutations, i.e. by figuring out which permutations result from performing two permutations successively. This will result in something that is analogous to a “multiplication table”. With all three properties shown to hold, the set of permutations of three elements forms a group, called the symmetric group $S_{3}$.

Although the definition of a group requires the law of composition to be associative, it does not require it to be commutative; for our example, two successive permutations might not give the same result when performed in the reverse order. When the law of composition of a group is commutative, the group is called an abelian group.

An example of an abelian group is provided by the integers, with the law of composition given by addition. Appropriately, we use the symbol $+$ to denote this law of composition. The identity element is provided by $0$, and the inverse of an integer $n$ is provided by the integer $-n$. We already know from basic arithmetic that addition is both associative and commutative, so this guarantees that under addition the integers form a group and moreover form an abelian group (sometimes called the additive group of integers).

That’s it for now, but the reader is encouraged to explore more about groups since the concept can be found essentially everywhere in mathematics. For example, the positive real numbers form a group under multiplication. The reader might want to check if they really do satisfy the three properties required for a set to form a group. Another thing to think about is the group of permutations of the set with three elements, and how they relate to the symmetries of an equilateral triangle. Once again the book of Artin provides a very reliable technical discussion of groups, but one more accessible book that stands out in its discussion of groups is Love and Math: The Heart of Hidden Reality by Edward Frenkel, which is part exposition and part autobiography. The connections between groups, symmetries, and physics are extensively explored in that book, as the author’s research explores the connection between quantum mechanics and the Langlands program, an active field of mathematical research where groups once again play a very important role. More on groups are planned for future posts on this blog.

References:

Groups on Wikipedia

Symmetric Group on Wikipedia

Dihedral Group on Wikipedia

Abelian Group on Wikipedia

Algebra by Michael Artin

Love and Math: The Heart of Hidden Reality by Edward Frenkel