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.