What Is Group Theory? The Mathematics of Symmetry

A group is a set of elements together with a way of combining any two of them, satisfying four conditions. Closure means combining two elements always gives another element of the set. Associativity means that when combining three, the grouping does not matter. There is an identity element, which leaves anything it is combined with unchanged. And every element has an inverse, another element that combines with it to give the identity. That is the whole definition, and its power comes from its emptiness: it says nothing about what the elements are, so anything satisfying it, whether rotations, whole numbers under addition, permutations of a list or matrices, obeys every theorem proved about groups in general. Notice what is absent: the definition does not require that combining two elements gives the same answer in either order, and groups where it does not are the interesting ones, since rotating then flipping a square is not the same as flipping then rotating.

Once the pattern is recognised it turns up constantly:

• •The symmetries of any shape, which is the original motivation, with the eight moves of a square and the six of an equilateral triangle as the standard first examples
• •Whole numbers under addition, where the identity is zero and the inverse of a number is its negative
• •Clock arithmetic, where the numbers 0 to 11 under addition wrap around, giving a finite group used throughout cryptography
• •Permutations, meaning all the ways of rearranging a set of objects, which form a group under composition and are central because every finite group can be seen as a group of permutations
• •Rotations in three dimensions, which describe the orientation of any rigid body and are used in robotics and computer graphics
• •The Rubik's cube, whose legal moves form a group with about 43 quintillion elements, and whose solution methods are group theory in practice

The subject came out of a question about equations. Formulas exist to solve any quadratic, and sixteenth-century Italian mathematicians found formulas for cubics and quartics, so the natural expectation was that a formula for the fifth degree would follow. It did not, for three centuries. Niels Henrik Abel proved in 1824 that no general formula in radicals exists, and Evariste Galois, working at nineteen and twenty, showed why, by associating to each equation a group describing how its solutions can be permuted without disturbing the relationships between them, and demonstrating that the equation is solvable by a formula exactly when that group has a particular structure. The groups attached to general fifth-degree equations do not have it. Galois wrote much of this out in a long letter the night before he was killed in a duel in 1832 at twenty, and it was not understood or published for more than a decade. The method, now called Galois theory, also settled several ancient geometry problems, proving that an arbitrary angle cannot be trisected with compass and straightedge and that a cube cannot be doubled.

Groups can be built from simpler ones, and the atoms of the construction are the finite simple groups, which cannot be broken down further. Classifying every one of them became one of the largest collaborative projects in the history of mathematics, occupying about a hundred mathematicians across roughly fifty years and producing a proof spread over hundreds of papers and tens of thousands of pages, announced complete around 1983 with gaps found and filled afterwards. The answer is that every finite simple group falls into one of a small number of infinite families, plus exactly twenty-six exceptions called sporadic groups that belong to no family. The largest sporadic group, the Monster, has roughly 8 times 10 to the 53rd elements, and an unexpected numerical coincidence between its properties and an apparently unrelated area of number theory, noticed in the late 1970s and named monstrous moonshine, turned out to be a genuine deep connection whose explanation drew on string theory and won a Fields Medal.

In physics the link is made by a theorem proved by Emmy Noether in 1918, which says that every continuous symmetry of a physical system corresponds to a conserved quantity. Symmetry under translation in space gives conservation of momentum, under translation in time gives conservation of energy, and under rotation gives conservation of angular momentum. The standard model of particle physics is built from particular symmetry groups, and the particles themselves are classified by how they transform under them, which is how Murray Gell-Mann predicted an undiscovered particle in 1962 from a gap in a symmetry pattern, and it was found two years later. In chemistry, the symmetry group of a molecule determines which of its vibrations absorb infrared light and which are visible in Raman spectroscopy, so a spectrum can be predicted from shape alone. In crystallography, group theory proves there are exactly 230 possible three-dimensional crystal symmetry types and exactly 17 possible repeating wallpaper patterns, all of which appear in the tilework of the Alhambra.

A group is any collection of operations that can be combined, where combining stays inside the collection, there is a do-nothing operation, and everything can be undone. The definition says nothing about what the elements are, so theorems about groups apply to symmetries, numbers, permutations and rotations alike. Galois invented the subject to explain why no formula solves the general fifth-degree equation, the finite simple groups were classified by a fifty-year collaborative proof, and Noether's theorem ties every continuous symmetry to a conservation law.