Beneath the Surface of Mathematics: An Introduction to Group Theory
Establishing the mathematical foundations for group theory and abstract algebra
Preface
Mathematics, at its heart, revolves around numbers and thus, the notion of counting: counting the number of numbers that satisfy specific properties; counting the lengths and angles of geometric figures; counting and manipulating the known in order to determine an unknown.
Numbers were created in order to assist in solving various problems involving counting. This began with the creation of the natural numbers, denoted as ℕ, which are all the positive whole numbers. And these satisfied our immediate mathematical needs. Eventually, these were extended into the integers, ℤ, which encompass the positive and negative numbers. This was then extended into the rational numbers, ℚ, then to the real numbers, ℝ. And this was where our number system stood until the introduction of complex numbers, ℂ, which extend the real number line via the addition of a second dimension. Each extension to the original number system served to satisfy additional needs and solve increasingly sophisticated problems. When complex numbers were formalized, it catalyzed a paradigm shift, with mathematicians beginning to realize that maybe the real numbers weren’t the ultimate system.
Entirely new number systems were created, which were used for solving an array of mathematical and practical problems that would largely be inapproachable if we were constrained to the real or complex number system. These include the creation of the p-adic numbers, quaternions, and Galois fields. With the creation of these number systems, mathematics strived to discover a unifying theory — a set of rules that governs these systems and outlines their behaviour — which led to the creation of abstract algebra. Abstract algebra is the study of algebraic structures. Algebraic structures comprise a non-empty set, known as the underlying set; a collection of mathematical operations with finite arity on the underlying set; a finite list of axioms that these operations must satisfy. More intuitively, abstract algebra can be thought of as the generalized study of number systems, studying what happens when certain properties and notions are abstracted out. There are various algebraic structures that arose, with various applications, but one of the most simple, yet powerful, was that of groups.
Group Theory
A group consists of a set and an operation. This operation must satisfy the property that it combines two elements of the set in order to produce a third element of the set — this is known as closure — , such that the operation is associative, there exists an identity element, and every element has an inverse. What does this actually mean? Let * denote a binary operation and
a, b, c members of the group’s set. The property of associativity means that
(a * b) * c = a * (b * c) i.e. the order in which we perform the calculations is irrelevant. The property of commutativity (i.e. the order of operands is irrelevant to the result of the operation) is often discussed in conjunction with associativity. If a group’s operation does satisfy the commutative property, then the group is said to be abelian, but this is not a requirement. An identity element of the binary operation is an element e such that a * e = a, for all possible values of a. Note that a group can only have one identity element i.e. there is exactly one value for e. Finally, if we have a, b such that a * b = e, then we say that a is the left inverse of b and b is the right inverse of a. Since the operation is associative by definition, then the left and right inverses are unique and equivalent; thus the inverse of an element is its unique left or right inverse. The inverse of an element is denoted by a -1 superscript, so b = a⁻¹. It should be noted that operations can be chained together (as a result of the closure axiom); when we perform an operation n times on an object a, it is denoted as aⁿ and can be treated using the properties of exponents. However, this does not mean that the operation is that of multiplication; it simply denotes an operation that is performed upon an element nth-fold. And with the essential notation established, we can conclude our discussion on the axioms of group theory.
The axioms stated are the weakest (read: most fundamental), from which all other theorems can be derived. For instance, we find that the properties of the identity and inverse are commutative (a * e = e * a; a * a⁻¹ = a⁻¹ * a), with the proof of these left as an exercise to the reader, following tradition. The focus of this article isn’t to get bogged down by the axiomatic details but to capture the essence of group theory and its fundamental notions. Axioms are at the heart of mathematics and provide us to understand groups at their core. Groups can essentially be thought of as a set that has been enriched by the addition of a mathematical operation. For instance, the set of integers, ℤ, coupled with the operation of addition, +, creates the group (ℤ, +). However, (ℤ, ×) is not a valid group, as the inverse property is not satisfied (the result being a rational number of the form 1/a, which is not a part of the underlying set).
The number of elements in a group is known as the order of the group and is represented as |(S, *)|. If |S| is infinite, then the group's order will also be infinite. However, if we were to look at the cyclic, multiplicative group of integers modulo n. (ℤₙ, *), then the order is n. However, groups needn’t be restricted to just the field of number theory; the symmetry group of a geometric object outlines all of the transformations under which the object is invariant (i.e. after undergoing a transformation, the object remains unchanged). More concretely, the dihedral group of a regular polygon specifies its symmetries, including its rotations and reflections. Dₙ refers to the symmetries of the n-th polygon, with |Dₙ| = 2n (there are n rotational symmetries and n reflection symmetries). It is interesting to note that commutativity is not a property of a dihedral group; commutativity is hence not a requirement of a group.
So far, we’ve gone through the formal definition of a group, its properties, and various examples. Group theory shows us how the sausage gets made for number systems, but its reach extends far beyond just pure numbers and arithmetic operations. However, these foundations are still largely abstract; to see why group theory matters, let’s peer a bit deeper into its depths.
Subgroups and Cosets
Subgroups are, as one might expect, a section of a larger group which happens to be a group itself (i.e. satisfies the necessary axioms). For instance, the even integers endowed with the operation of addition are a subgroup of (ℤ, +); it is interesting to note, however, that the odd integers are not a subgroup, due to the fact that they do not satisfy the closure property — an odd integer added to another odd integers results in an even integer, which is not contained within the underlying set.
The concept of subgroups arises naturally when talking about a closed system (i.e. a group wherein the underlying set has finite cardinality). When we discuss the set of integers modulus a number, or the set of symmetry-preserving rotations of a polygon, we are talking about a closed system. In general, for any element a of a finite group, the subgroup generated by a (denoted as ⟨a⟩) is defined as the multiples of a: ⟨a⟩ = {a, a², a³, …}. As this subgroup is finite, we will eventually run out of multiples of a (a subgroup’s order must be less than or equal to its supergroup). This group satisfies the property of closure due to the laws of exponents. It also satisfied the identity axiom, as eventually, we will have no choice but to create an identity element as a power of a (this can be thought of as a ‘modulus’ of the group and is the greatest achievable multiple of a). We easily find that the inverse axiom is also satisfied, with associativity preserved from the original group. We’ve now demonstrated that the subgroup is, indeed, a group, as well as witnessed the cyclical nature that arises thanks to the properties of a closed system. It is important to note that this notion of a subgroup was developed for a finite group; the above process can fail horribly when attempting to replicate it for an infinite group. Subgroups allow us to gain a deeper understanding of the properties of groups themselves, as it turns out that groups are highly symmetric with respect to their subgroups. This symmetry arises from the uncanny prevalence of shifted copies of subgroups or, mathematically speaking, cosets.
A coset is nothing more than the result of shifting every element in the underlying set of a group by a finite amount. More formally, if H is a subgroup of G, endowed with an operation *, then a coset is generated by operating every element of H with a fixed element of G. For instance, by shifting the subgroup of the even integers to the right by 1 (i.e. adding 1 to every element of the subgroup), we achieve the coset of all the odd integers. In the above example, we would denote the coset as 1+⟨2⟩. Note that a coset does not necessarily have to be a group. We can find similar symmetries amongst the dihedral group, where the subgroup of rotations is shifted by a flip. Through cosets, we can see how a group can be neatly divided and organized, with elements unique and ordered.
I previously mentioned that groups have an innate symmetry present with respect to their subgroups. We can now quantify this notion exactly. Suppose we have group G with subgroup H. Then, the cosets of H neatly partition the group into equally-sized sections. As a result, the cosets cover the group in its entirety and each coset must be of equal size to the subgroup. These are easily justifiable: in order to cover any specific element of G, x, simply shift the group by x as x * e = x; each coset must be of identical order, as during a shift, an element cannot be gained (as the result of an operation is a single result) or lost (as if this was the case, then a single element must have two distinct inverses in order to shift back, but we have previously shown this to be impossible). The final observation, which is less obvious at first glance, is that distinct cosets will never overlap. Let’s say we shift the subgroup by a∈ H. As a result of closure (a * a = b∈ H), the shift will be contained within the subgroup and no new coset is created. Instead, suppose we shift the subgroup by z∉ H and that z* a = b∈ H. Applying the inverse of a, we get z = b * a⁻¹. However, since both b and a⁻¹ (by definition, if a is an element then so must its inverse) are elements of the subgroup, z must also be an element by closure. However, as we stated that z was outside of the subgroup, its result could never have been within the subgroup, to begin with.
As such, we conclude that a coset either does not overlap with the original subgroup at all, or it is the subgroup itself. This argument can be repeated between cosets, and as a result, we can prove the original group will be evenly partitioned by its cosets. This is best summarized by Lagrange’s Theorem: for a finite group G with subgroup H, |G| is divisible by |H|. Why does this matter? Think about the applications of this theorem to a group with prime order p. Since the only factors of p are itself and one, a subgroup may only have either 1 or p elements. Now, if we take any element which is not the identity and start generating its subgroup, that subgroup must be of order p, so after reaching the neutral element, we will have filled the whole group. As such, any group, no matter how abstract it may be, of prime order p must be a cyclic group.
Applications & Conclusion
The question of why is generally the most difficult to answer. Groups provide an abstraction for number systems and enable us to gain a deeper insight into their properties. Within mathematics, they find their place within Galois theory — the bridge between field theory and group theory — , algebraic geometry — which is the study of roots of multivariate polynomials, and harmonic analysis — which is the branch of mathematics that uses the superposition of waves to represent functions. However, we find that groups are quite useful and are applied in areas as diverse as music, physics, molecular science, and cryptography. Group theory is a vast and fascinating field of mathematics and is just the tip of the iceberg that is abstract algebra. The power of the abstract is its ability to be broadly applicable to practicality, allowing us to derive unique insights and solve problems in entirely novel ways. And group theory is a prime example of that, as well as of the rich world that lies beneath the surface of mathematics.
