Broadly speaking, group theory is the study of symmetry. When we are dealing with an object that appears symmetric, group theory can help with the analysis. We apply the label symmetric to anything which stays invariant under some transformations. This could apply to geometric figures (a circle is highly symmetric, being invariant under any rotation), but also to more abstract objects like functions: x² + y² + z² is invariant under any rearrangement of x, y, and z and the trigonometric functions sin(t) and cos(t) are invariant when we replace t with t+2π.

Conservation laws of physics are related to the symmetry of physical laws under various transformations. For instance, we expect the laws of physics to be unchanging in time. This is an invariance under "translation" in time, and it leads to the conservation of energy. Physical laws also should not depend on where you are in the universe. Such invariance of physical laws under "translation" in space leads to conservation of momentum. Invariance of physical laws under (suitable) rotations leads to conservation of angular momentum. A general theorem that explains how conservation laws of a physical system must arise from its symmetries is due to Emmy Noether.

Modern particle physics would not exist without group theory; in fact, group theory predicted the existence of many elementary particles before they were found experimentally.

The structure and behavior of molecules and crystals depends on their different symmetries. Thus, group theory is an essential tool in some areas of chemistry.

Within mathematics itself, group theory is very closely linked to symmetry in geometry. In the Euclidean plane R², the most symmetric kind of polygon is a regular polygon. We all know that for any n > 2, there is a regular polygon with n sides: the equilateral triangle for n = 3, the square for n = 4, the regular pentagon for n = 5, and so on. What are the possible regular polyhedra (like a regular pyramid and cube) in R³ and, to use a more encompassing term, regular "polytopes" in R^d for d > 3?

• In R³, there are only five (convex) regular polyhedra, called the Platonic solids:

The Five Platonic Solids

• In R⁴, there are only six (convex) regular polytopes.
• For any d > 4, the number of (convex) regular polytopes in R^d is always three: the higher-dimensional analogues of the tetrahedron, cube, and octahedron.

Consider another geometric topic: regular tilings of the plane. This means a tiling of the plane by copies of congruent regular polygons, with no overlaps except along the boundaries of the polygons. For instance, a standard sheet of graph paper illustrates a regular tiling of R² by squares (with 4 meeting at each vertex).

Tiling the Plane with Congruent Squares

There are also regular tilings of R² by equilateral triangles (with 6 meeting at each vertex) and by regular hexagons (with 3 meeting at each vertex).

Tiling the Plane with Congruent Equilateral Triangles

Tiling the Plane with Congruent Regular Hexagons

But that is all: there are no tilings of R² by (congruent) regular n-gons except when n = 3, 4, and 6 (e.g., how could regular pentagons meet around a point without overlap?). Furthermore, for these three values of n there is essentially just one regular tiling in each case. The situation is different if we work with regular polygons in the hyperbolic plane H², rather than the Euclidean plane R². The hyperbolic plane H² is the interior of a disc in which the "lines" are diameters passing through the center of the disc or pieces of circular arcs inside the disc that meet the boundary of the disc in 90 degree angles. See the figure below.

Lines in the Hyperbolic Plane H2

In H² there are tilings by (congruent) regular n-gons for any value of n > 2! Here is a tiling of H² by congruent regular pentagons.

Tiling the Hyperbolic Plane with Congruent Regular Pentagons

The figures here are pentagons because their boundaries consists of five hyperbolic line segments (intervals along a circular arc meeting the boundary at 90 degree angles). The boundary parts of each pentagon all have the same hyperbolic length (certainly not the same Euclidean length!), so they are regular pentagons, and the pentagons are congruent to each other in H² even though they don't appear congruent as figures in the Euclidean plane. Thus we have a tiling of H² by congruent regular pentagons with four meeting at each vertex. Nothing like this is possible for tilings of R², the Euclidean plane. There is also more than one tiling of H² by regular n-gons for the same n. For instance, taking n = 3, there is no tiling of H² by congruent equilateral triangles meeting 6 at a vertex, but there are tilings of H² with 7 meeting at a vertex and 8 meeting at a vertex. Here they are.

Tiling the Hyperbolic Plane with Congruent Equilateral Triangles, 7 at a Vertex

Tiling the Hyperbolic Plane with Congruent Equilateral Triangles, 8 at a Vertex

What makes it possible to tile H² by regular polygons in more possible ways than R² is the different structure of the group of rigid motions (distance-preserving transformations) in H² compared to R². The German mathematician Felix Klein gave a famous lecture in Erlangen, in which he asserted that the definition of a geometry is the study of the properties of a space that are invariant under a chosen group of transformations of that space.

Group theory shows up in many other areas of geometry. For instance, in addition to attaching numerical invariants to a space (such as its dimension, which is just a number) there is the possibility of introducing algebraic invariants of a space. That is, one can attach to a space certain algebraic systems. Examples include different kinds of groups, such as the fundamental group of a space. A plane with one point removed has a commutative fundamental group, while a plane with two points removed has a noncommutative fundamental group. In higher dimensions, where we can't directly visualize spaces that are of interest, mathematicians often rely on algebraic invariants like the fundamental group to help us verify that two spaces are not the same.

Classical problems in algebra have been resolved with group theory. In the Renaissance, mathematicians found analogues of the quadratic formula for roots of general polynomials of degree 3 and 4. Like the quadratic formula, the cubic and quartic formulas express the roots of all polynomials of degree 3 and 4 in terms of the coefficients of the polynomials and root extractions (square roots, cube roots, and fourth roots). The search for an analogue of the quadratic formula for the roots of all polynomials of degree 5 or higher was unsuccessful. In the 19th century, the reason for the failure to find such general formulas was explained by a subtle algebraic symmetry in the roots of a polynomial discovered by Evariste Galois. He found a way to attach a finite group to each polynomial f(x), and there is an analogue of the quadratic formula for all the roots of f(x) exactly when the group associated to f(x) satisfies a certain technical condition that is too complicated to explain here. Not all groups satisfy the technical condition, and by this method Galois could give explicit examples of fifth degree polynomials, such as x⁵ - x - 1, whose roots can't be described by anything like the quadratic formula. Learning about this application of group theory to formulas for roots of polynomials would be a suitable subject for a second course in abstract algebra.

The mathematics of public-key cryptography uses a lot of group theory. Different cryptosystems use different groups, such as the group of units in modular arithmetic and the group of rational points on elliptic curves over a finite field. This use of group theory derives not from the "symmetry" perspective, but from the efficiency or difficulty of carrying out certain computations in the groups. Other public-key cryptosystems use other algebraic structures, such as lattices.

Some areas of analysis (the mathematical developments coming from calculus) involve group theory. The subject of Fourier series is concerned with expanding a fairly general 2π-periodic function as an infinite series in the special 2π-periodic functions 1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), and so on. While it can be developed solely as a topic within analysis (and at first it was), the modern viewpoint of Fourier analysis treats it as a fusion of analysis, linear algebra, and group theory.

Identification numbers are all around us, such as the ISBN number for a book, the VIN (Vehicle Identification Number) for your car, or the bar code on a UPS package. What makes them useful is their check digit, which helps catch errors when communicating the identification number over the phone or the internet or with a scanner. The different recipes for constructing a check digit from another string of numbers are based on group theory. Usually the group theory is trivial, just addition or multiplication in modular arithmetic. However, a more clever use of other groups leads to a check-digit construction which catches more of the most common types of communication errors. The key idea is to use a noncommutative group.

On the lighter side, there are applications of group theory to puzzles, such as the 15-puzzle and Rubik's Cube. Group theory provides the conceptual framework for solving such puzzles. To be fair, you can learn an algorithm for solving Rubik's cube without knowing group theory (consider this 7-year old cubist), just as you can learn how to drive a car without knowing automotive mechanics. Of course, if you want to understand how a car works then you need to know what is really going on under the hood. Group theory (symmetric groups, conjugations, commutators, and semi-direct products) is what you find under the hood of Rubik's cube.