Group (Abstract Algebra)

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

A square looks the same after a quarter turn, a half turn, or a flip along either diagonal; a circle looks the same after a turn of any size at all. Both statements describe symmetry, and for most of mathematical history symmetry was treated as a property an object happens to have, described case by case, rather than as a mathematical object in its own right. The group is what changed that: take every way of moving an object that leaves it looking unchanged, and it turns out those moves themselves combine according to a small, fixed set of rules, closure, associativity, an identity move that does nothing, and an inverse undoing every move, the same four rules regardless of whether the object being moved is a square, a circle, or, as Evariste Galois discovered first, the roots of a polynomial equation. Once symmetry itself became an object that could be studied on its own terms, the same structure turned out to describe an enormous range of things that do not look like symmetry at first glance: the ways a Rubik's cube can be scrambled, the conserved quantities in particle physics, the error-correcting codes that let a scratched CD still play. Arthur Cayley's 1854 papers were the first attempt at an abstract definition of a group, detached from any one example, but the attempt was so far ahead of its time that it had little impact at first; it was only when Cayley returned to the subject in 1878, and mathematicians including von Dyck, Weber and Burnside built on that return over the following two decades, that mathematicians began recognizing the same underlying structure in problems that had nothing else in common.

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

The word group in mathematics means something so general that the ordinariness of the English word is almost the point. Evariste Galois used a French equivalent of the word informally in his own manuscripts to describe collections of permutations he was studying, but it was Arthur Cayley, in an 1854 paper, who first attempted a definition in the abstract, applicable to any set with any operation that happened to satisfy it, whether or not the elements had anything to do with equations at all. That 1854 definition covered three of what are now considered the four defining properties, closure, associativity and an identity element, but left out inverses; the clean, complete four property statement came only when Cayley returned to the subject in 1878. That deliberate emptiness is exactly what has made the concept so durable: because a group is defined only by how its elements combine, not by what the elements themselves are, the same theorems proved once about groups in general apply automatically to the integers under addition, to the rotations of a molecule, and to the moves of a puzzle, without needing to be reproven in each separate case. It was in that 1878 return that Cayley went further and showed that every finite group, however it is defined, can be represented concretely as a group of permutations, a result now named for him: whatever abstract structure a group has, it can always be realized as symmetries acting on some set of objects. It is a small, quiet piece of unification, one definition doing the work that used to require a separate argument for every different kind of symmetry mathematics happened to run into.