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.