Group theory is the language of many of the mathematical disciplines. An indispensable tool in understanding the underlying nature of nature.

A theory that holds the secrets of the fundamental particles and forces of the Universe itself!

We use it to understand shapes in higher dimensions, the proof of insolvability of higher degree polynomials, and the structure of number systems, but the use of this beautiful theory doesn’t stop there.

Group theory is the study of symmetry and as you will soon see, symmetry is everywhere in mathematics and in nature.

My own encounter with groups started when I studied mathematics at University.

Before then, even though it was quite early in my studies, I had had a bunch of courses in different kinds of analytic disciplines, e.g. real analysis, complex analysis, harmonic analysis, Fourier series, measure theory, Hilbert space theory, etc…

That was all great but at that time, I had the impression that there wasn’t much more to mathematics than differentiation, different kinds of integration theories, infinite series, transformations of different spaces, differential equations, and the likes of it.

Suffices to say that I was very wrong!

My first course in abstract algebra was about this thing that they called a group.

I was instantly in love. And that was even before I got to know of all the applications of groups to other mathematical disciplines and sciences.

I was in love with the pure abstract universe of groups.

The subject itself was beautiful and I was fortunate enough to have some great professors and TAs helping us getting through this exotic world, but I still felt like “there must be another way of teaching someone this subject”.

I personally saw the beauty but not all did and many drowned and flunked the introductory courses in abstract algebra because of the abstractness and difficulty of this subject.

I will try to teach this subject in a way that I think I would have liked when I first studied it.

I will blend theory with examples so that the reader quickly builds up an intuition about groups.

Into Abstractness — the Beginning of Algebra

When first starting out on this journey, the first thing you should do is to forget all the things that you think you know.

If the cup is full, you can’t pour water into it and if you assume too many things in group theory, you will get your fingers burnt because when it comes to groups, a lot of your intuition about numbers cannot be used.

You don’t know more than you can prove - so be careful.

The Axioms

It is tradition to define a group in terms of a set of properties called “axioms”. I think that the naming convention here is a little unfortunate in that it has nothing to do with the ZFC-axioms or set-theoretic aspects of mathematical logic or anything of that nature.

We just call the properties that a group needs to satisfy, “axioms”. It will become clearer why we do this a little later though.

By the notation “a ∈ G”, we mean “a is an element of the group G”.

Let me state the axioms and then immediately give you some examples in order to demystify and decomplexify the structure before actually proving some things about groups.

Definition:

A group consists of a set G and a binary operation ⋅ (not necessarily multiplication) on G satisfying the following “axioms”:

• If a, b ∈ G then a ⋅ b ∈ G (i.e. the group is closed with respect to the group operation)
• (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) for all a, b, c ∈ G (i.e. the group operation is associative)
• There exists an element e in G called an identity of G, such that for all a ∈ Gwe have a ⋅ e = e ⋅ a = a
• For each a ∈ G there exists an element d ∈ G called an inverse of a such that a⋅ d = d ⋅ a = e.

Sometimes we denote this structure (G, ⋅).

Examples:

• The whole numbers ℤ = {…, -2, -1, 0, 1, 2, …} is a group with respect to the operation + and of course with 0 as the identity element. Notice that every element n ∈ ℤ has an inverse element, namely -n because n + (-n) = n - n = 0. We denote this (ℤ, +).
• The group of non-zero rational numbers (ℚ\{0}, ⋅) which consists of all fractions of whole numbers (except for 0 in the denominator and numerator of course) is a group with respect to the operation of multiplication with 1 as the identity element. Notice that this group has an inverse element to any given element n/m namely m/n, because n/m ⋅ m/n = 1.
• n × n invertible matrices form a group (called the general linear group) with respect to matrix multiplication. We denote this GL(n, ℝ).
• The actions of rotation by 120 degrees and reflections about the three obvious axes on an equilateral triangle is a (finite) group. Note that there are exactly 6 symmetries of an equilateral triangle.

Notice that all these groups have a symmetric nature.

The whole numbers have a 1-dimensional symmetry around 0, its identity element. Each element’s inverse element lies symmetrically (by reflection) on the opposite side of 0.

If you display the non-zero rational numbers as points in the plane ℤ²\(0, 0) so that e.g. 3/2 corresponds to the point (3, 2), 1/2 corresponds to the point (1, 2), etc. then there is a 2-dimensional symmetry along the diagonal in the sense that a reflection through that diagonal line maps a point to its inverse point just like the symmetry of ℤ. And the symmetry line is what is known as the equivalence class containing the number 1, the identity element of ℚ.

Displaying the fractions like this is not far-fetched - the set of rational numbers actually consists of equivalence classes — as you will see soon, a group has one and only one identity element so what is up with the two fractions 1/1 and 2/2? They are both equal to 1 of course, but we need to formalize what we mean by a rational number then.

Without getting into a long discussion about mathematical relations on sets, I can tell you that the solution to this is in terms of a special kind of relation called an equivalence relation.

Of course, the triangle has obvious symmetries.

The general linear group (the matrix group above) is also about symmetries but that is a little harder to explain.

Properties of Groups
Let’s prove some fundamental properties of groups from first principles.

These facts are great small exercises. If you want to give it a try yourself before reading the proof, you are more than welcome.

Just remember: You are only allowed to use the axioms of a group in your arguments. When you read the proofs, try to be as sceptic as possible. See if you can pinpoint exactly which axiom is used where.

Proposition 1: In a group, the identity element is unique.

Proof: Suppose that e and a are both identity elements of G, then a⋅ e = a. Likewise, we get that a ⋅ e = e. This implies that a = e, thus there is only one identity in a group.

Q.E.D.

If we consider the whole numbers as a group we knew that the only whole number z with the property that z + n = n is z = 0, making the identity element for the whole numbers unique, but this is actually true for all groups!

Proposition 2: In groups, inverses are unique.

Proof: Let b, c ∈ G be inverse elements of an element a ∈ G and let e be the identity of G. Then we have the following:

c= c ⋅ e = c ⋅ (a ⋅ b) = (c ⋅ a) ⋅ b = e ⋅ b = b.

Q.E.D.

Notice how we used the axioms in the proof. The property of the identity, the associativity, and the inverse property.

Moreover, note that the axioms state nowhere anything about commutativity, and in fact, in general in a group we have a ⋅ b ≠ b ⋅ a. This is, for example, true in both the matrix group above and in the symmetries in a triangle e.g. a rotation followed by a reflection is not the same as first doing a reflection and then a rotation.

Homomorphisms

Before deep-diving into the world of groups and mappings between them, there is one important discussion that we need to have.

The study of sets in mathematics is an important one. In fact, all of mathematics relies on a handful of axioms from set-theory called the ZFC-axioms. In that way, in principle, it should be possible to distill a proof down to axioms from set theory! It is the concrete foundation on which all other mathematics is built.

It hasn’t always been this way, but that is for another story.

But what are sets really and what properties do they possess?

Well, besides being able to contain other sets, (numbers are considered sets in this context), they have only one interesting characteristic, their cardinality (i.e. how many elements they contain).

A set has no other structure by default inside the set. Only what it contains and how many elements it contains.

For example, the set S = {1, 2, 3} is a fine set, but we need to be careful when we think about what it contains. Obviously, it contains three numbers, so the cardinality is 3. Then, most people (including mathematicians) will say that the element 1 ∈ S is less than the element 2 ∈ S.

Here’s the thing… Sets don’t come with orderings by default. We can create an ordered mathematical structure, by imposing an order on a set - a relation to be more formal. And this relation can also be stated in terms of sets of course, but if that is not given, then the set S is just a set - nothing else.

So in fact, unless we state that the set S is a subset of some set with an ordering (like the natural numbers f.x.), or that it comes with some order, we cannot say that one element is greater than the other).

So really, the study of sets themselves is about cardinality.

It turns out that that the right tool for studying and comparing sets are functions. It also turns out that certain types of functions preserve the sizes of sets, which is of huge interest to us.

They are called bijections, and they are very important. They can be seen as structure-preserving as well as cardinality preserving mappings between sets.

Georg Cantor was the first person to realize this by the end of the 19’th century and now it has become standard practice.

Many times in mathematics, to study a certain kind of structure, it can be beneficial to study the structure preserving mappings between them.

This is true for sets, groups, vector spaces, rings, modules, topological spaces, etc. and the generalized study of these mappings is called category theory.

So the natural kind of mappings to study in set theory are functions between sets since sets do not really have structure.

By the same token, the structure-preserving mappings between ordered sets are monotonic functions.

What is the structure preserving maps between groups?

It turns out that the natural structure-preserving map to study in group theory is a special kind of function called a homomorphism.

A homomorphism is a function f between the underlying sets of two groups say G and H i.e. f: G → H, sending elements of G to elements of H but with an important restriction, namely that it always holds that f(a ⋅ b) = f(a) * f(b) where ⋅ is the group operation in G and * is the group operation in H.

In that way, we can translate structure in G to structure in H.

Examples

Consider the group G with respect to (complex) multiplication G = ({1, -1, i, -i}, ⋅). Here i is the imaginary unit i.e. i² = -1.

There is a homomorphism between G and the group of integers modulo 4 with respect to addition. We denote the latter group (ℤ/4ℤ, +).

The mapping f: (ℤ/4ℤ, +)→ G given by

• 0 → 1
• 1 → i
• 2 → -1
• 3 → -i

is a homomorphism between the two groups. In fact, more is true.

When a homomorphism is also a bijection (one-to-one and onto) then it is called an isomorphism (meaning: same shape).

If there exists an isomorphism between two groups it means that these are the same mathematical objects up to naming. So really the same thing. The groups are then called isomorphic.

Actually, the homomorphism above between the complex numbers called the group of units of the Gaussian numbers and the group of integers mod 4 with respect to addition is an isomorphism, showing that the two structures are equivalent.

That it is a bijection is trivial.

You can easily check that it is a homomorphism since there are only finitely many cases to check. One of them is:

f(2 + 3) = f(1) = i = -1 ⋅ (-i) = f(2) ⋅ f(3).

The isomorphism is no surprise since it is well-known that you can view complex numbers as implementing a rotation transformation on the real numbers and extend the real line to the complex plane. The element responsible for rotating 90 degrees counter-clockwise while keeping the distance to the origin is, of course, the imaginary unit i, and since (ℤ/4ℤ, +) is a cyclic group generated by the element 1, we can translate addition in this group into rotations on the unit circle in the complex plane (a little like an hour hand on a watch with only 4 hours).

You can read more about symmetries in numbers here:

The Most Beautiful Equation in the WorldAnd the Geometry of Numbersmedium.com

This shows that modular arithmetic is really about winding around a clock. Interesting!…

Another example is the well-known function exp(x) sometimes denoted e^x.

This is in fact a homomorphism from the additive group of real numbers (ℝ, +) to the multiplicative group of non-zero real numbers (ℝ*, ⋅).

Clearly exp(x+y) = exp(x) ⋅ exp(y).

That’s why we study the exponential function!

That’s why it is so useful in solving differential equations. It is a group homomorphism! There is a lot more to be said about the exponential function as a group homomorphism. If we consider it as a function of complex numbers it has complex periodicity and gives all the symmetries of the unit circle which happens to be a Lie group but I better stop here…

Subgroups

A natural concept when one studies mathematical objects, in general, is that of a subobject. We have subspaces of vector spaces, subsets of sets, and so on and so forth.

Therefore it is no surprise that we, in group theory, have the concept of a subgroup. A subgroup is a group in which the corresponding elements that it consists of form a subset of the set of elements of the group.

An example of this is the group of even integers with respect to the operation of addition (2ℤ, +). You can verify that this is indeed a group that is of course a subset of the (group of) whole numbers ℤ.

We denote this (2ℤ, +) < (ℤ, +), and in general, if H is a subgroup of G, we write H < G.

It turns out that a subspace of a vector space actually is a subgroup and this is the case in many scenarios in mathematics.

Sometimes we need to study subgroups and homomorphisms in order to understand the group itself. For example, in linear algebra, we talk about the column space and the row space of a matrix and translate that into the solvability of a system of linear equations in which the determinant of the coefficient matrix plays an important role. Guess what…. All this can be generalized by group theory.

• First of all, matrices as linear operators are actually homomorphisms between vector spaces.
• Secondly, the null space is (in group theory) what is called the kernel of the homomorphism, which we will talk about in the next article.
• The determinant is itself a homomorphism between the space of invertible matrices and the group of units of the underlying field (e.g. the positive real numbers).

Group theory lets us study a generalized form of linear algebra. It lets us realize universal truths about several different structures at once.

Number sets and solution sets (groups) of diophantine equations in number theory, matrices and vector spaces in vector analysis, permutation groups in combinatorics, symmetry groups in different dimensions for particle physics, and the standard model, characterizations of topological spaces in algebraic topology etc.

As for my own studies, I quickly realized that not only was group theory beautiful on its own, it was useful in other areas of mathematics.

After I studied more advanced topics like ring theory (abstractions of number fields which all have an additive structure of a group), module theory (kind of generalizations of abstract vector spaces over rings instead of fields), Galois theory (fields, permutation groups, and the bridge to polynomial roots), K-theory, Ideal theory, etc… I thought that I moved away from the real world a little. It became very abstract!

Then I had courses in algebraic topology and all of a sudden, all that theory paid off. It turns out that group theory is the best bet in understanding topological spaces of higher dimensional objects and that homomorphism between groups representing topological spaces can (in a very natural way) be interpreted as structure-preserving transformations of the spaces themselves.

So we had a dictionary between two seemingly different universes (topological and algebraic) and facts in one universe could be translated to facts in the other.

It was a match made in heaven. Understanding the shapes and spaces came down to understanding the structures of the groups representing them and vice versa.

All of a sudden, I saw the light at the end of the abstract tunnel in the applications to other areas.

In the next article, we will dive further into subgroups and Lagrange’s Theorem that states the following fact:

Let H < G with G finite. Then |H| divides |G|.

In other words: The number of elements of a subgroup of a finite group divides the number of elements of the group.

See you soon.