Group theory is the branch of mathematics that answers the question, "What is symmetry?"
Insight1.1.
There are several ways of thinking about groups, all relying or various levels of rigor and assumptions of prior experience. Here are several:
Symmetric.
A group can be thought of as a collection of symmetries or transformations that can be combined in a consistent way. Common examples of symmetries include moves on a Rubik’s Cube, addition and subtraction of integers, and rotations and flips of polygons.
Set Theoretical.
A group is a set of elements together with a rule (function) for combining any two elements to get a third element in the set.
Abstractions of Abstractions.
Groups can also be thought of as abstractions of rings, which can in turn be thought of as abstractions of common number systems, such as the integers. Rings are algebraic objects with two operations (commonly addition and multiplication), whereas groups only require one operation. Thus all rings can be thought of as containing groups inside them.
Common first questions in most math settings are the staples "Why are these useful?" and "Where are these found?". Despite what some mathematicians might insist, these are reasonable and often essential questions to ask when encountering a new mathematical concept. Questions should always be encouraged, anyone who tells you differently is either unprepared or underpaid.
Question1.2.
Why are groups important? Where are they found?
Answer.
The short answer is almost everywhere. In many fields of mathematics, groups provide the abstract framework upon which large portions of the discipline are built. Listed below is a thourogh but nowhere near comprehensive list of fields of mathematics (and beyond) where groups make an appearence.
Number Theory and Cryptography.
As it turns out, much of elementary group theory is actually group theory in disguise: specifically, the study of modular arithmetic groups and multiplicative groups of units. Thus many cryptographic protocols, such as RSA and elliptic curve cryptography, rely on the structure of groups for their security.
Linear Algebra.
A vector space is essentially a field and a group working together, with elements of the group acting as vectors and elements of the field acting as scalers. Additionally, the concept of eigenvalues and eigenvectors is deeply connected to group theory: the action of a linear transformation can be studied using its eigenvalues and eigenvectors, which form cyclic subgroups under repeated application of the transformation.
Commutative Algebra.
In many ways commutative algebra is the study of modules, which are essentially abstractions of vector spaces: where a field is required for a vector space, only a ring is needed for a module. Thus groups are once again omnipresent, though they remain mostly hidden in the machinery of the module structure. 1
This author can personally attest to the importance of this connection. As someone who struggled with modules for quite some time, it was expanding my understanding of groups that finally enabled me to grasp the inner workings of modules.
Topology.
An essential concept in topology is that of the fundamental group, which describes the set of loops in a topological space up to continuous deformation, providing insight into the space’s structure.
Algebraic Geometry.
The set of points on an elliptic curve forms an abelian group, which is essential in number theory and cryptography.
Representation Theory.
Group representations are ways in which a group can act on vector spaces, providing yet another bridge between group theory and linear algebra.
Physics.
Groups describe symmetries in physical systems, such as the Lorentz group in special relativity and gauge groups in quantum field theory. In the realm of quantum mechanics, quantization of a classical system crucially involves understanding the Lie groups that act on the classical phase space and the unitary representations of these groups.
Analysis.
In functional analysis, groups of unitary or orthogonal matrices are crucial in studying linear operators on Hilbert spaces. In complex analysis, automorphism groups of Riemann surfaces are essential in studying the geometry of complex manifolds. Additionally, Möbius Transformations, the group of transformations of the complex plane that preserve angles and are used in the study of complex functions.
Knot Theory.
A fundamental concept in knot theory is the knot group, which is an invariant that can distinguish between different knots. The knot group is defined as the fundamental group of the knot complement, and is used to derive powerful knot invariants, such as the Alexander Polynomial, the Jones Polynomial, and linking number.
Algebraic Topology.
Homology groups are used to study the topological features of a space, such as holes and voids, in a more algebraic manner.
Chemistry.
The set of operations associated with the symmetry elements of a molecule constitute a group. Indeed, the majority of all knowledge concerning atoms, molecules, and solids has been derived from applications of group theory.
List1.3.Supplemental Group Theory Resources
If you find yourself looking for additional material for learning group theory, I strongly recommend the following excellent resources:
