On the night of May 30, 1832, a twenty-year-old French radical named Évariste Galois — about to fight a duel he was certain he would lose — sat at his desk and wrote, frantically, a long letter to a friend. The letter contained the foundations of what we now call group theory: the abstract algebra of symmetry. Galois died the next morning, shot in the abdomen, leaving the notes with margins crowded by the words je n'ai pas le temps — I have no time. The notes were ignored for fifteen years. When Joseph Liouville finally read them and understood what they contained, he found that Galois had not only invented a new branch of mathematics but had used it to settle a question that had stumped Europe for three hundred years: why does no general formula exist for solving polynomial equations of degree five or higher?
A group is a set G with a binary operation · satisfying four axioms: closure (a · b ∈ G for all a, b ∈ G); associativity ((a · b) · c = a · (b · c)); identity (there exists e such that e · a = a · e = a); inverses (every a has some a⁻¹ with a · a⁻¹ = e). The minimum is austere; the consequences are vast. Examples are everywhere: integers under addition (an abelian, infinite group); nonzero rationals under multiplication; rotations of a square (the dihedral group D₄, order 8); permutations of n objects (the symmetric group Sₙ, order n!); rigid motions of the plane; invertible matrices under multiplication (general linear groups); the transformations that leave a physical system unchanged. The classical Galois insight is that the symmetries of the roots of a polynomial form a group, and the polynomial is solvable by radicals if and only if its Galois group is solvable in a precise group-theoretic sense — and the symmetric group S₅ is not solvable, hence the impossibility of a general quintic formula. Group theory's internal landscape: subgroups, cosets, normal subgroups, quotient groups, homomorphisms, isomorphisms, simple groups, solvable groups. The classification of finite simple groups — completed in 2004 after a 130-year, ten-thousand-page collaborative effort — is the largest theorem proof in mathematical history. Lie groups — groups that are also smooth manifolds — capture continuous symmetries and are the natural language of physics: rotations in 3D form SO(3), Lorentz transformations form SO(3,1), the Standard Model's gauge group is SU(3) × SU(2) × U(1).
Crystallography classifies all possible crystal structures by their symmetry groups (230 space groups in three dimensions). Particle physics describes elementary particles as representations of gauge groups; the entire Standard Model is built around symmetry constraints. Cryptography — particularly elliptic-curve cryptography, the basis of modern TLS, Bitcoin, and most secure communication — relies on the difficulty of computing discrete logarithms in carefully-chosen groups. Public-key encryption is, at its core, group-theoretic. The Rubik's cube has a group of about 4.3 × 10¹⁹ states, and the minimum-moves problem ("God's number") was settled in 2010 — every position is solvable in at most 20 moves. Music theory uses the cyclic group ℤ/12 to describe transposition. Group theory is the mathematics of invariance, and the universe seems to like invariance.