A group is a set equipped with a binary operation satisfying four axioms: closure, associativity, identity, and invertibility. Groups are the fundamental objects of abstract algebra — they appear wherever symmetry or transformation is present, from the rotation of a cube to the particles of the Standard Model.
Imagine the six ways you can rotate an equilateral triangle and still have it look the same — rotate by 0°, 120°, or 240°. Notice: doing two rotations in a row still gives a valid rotation. There's a "do nothing" rotation. And every rotation can be undone. This collection of rotations, together with the rule for combining them, is a group.
A group is a set of things — numbers, rotations, permutations, symmetries — combined with an operation that obeys four rules:
Groups are the mathematical language of symmetry — and symmetry appears everywhere: in physics, chemistry, music, and art. The concept was introduced by Évariste Galois around 1830, in work he wrote the night before being killed in a duel at age 20. He used groups to solve — after centuries of searching — why there is no formula for the roots of a general degree-5 polynomial.
A group is an ordered pair (G, ★) where G is a set and ★ is a binary operation on G satisfying:
If additionally a★b = b★a for all a, b, the group is abelian (commutative), named after Niels Henrik Abel.
| Group | Set | Operation | Identity | Abelian? |
|---|---|---|---|---|
| ℤ | Integers | + | 0 | Yes |
| ℚ*, ℝ*, ℂ* | Nonzero numbers | × | 1 | Yes |
| ℤ/nℤ | Integers mod n | + | 0 | Yes |
| Sₙ | Permutations of n | Composition | Identity | n ≤ 2 only |
| GL(n,ℝ) | Invertible n×n matrices | Mult. | I | n ≥ 2: No |
If G is a finite group and H ≤ G is a subgroup, then |H| divides |G|. A group of prime order p has no proper subgroups — every non-identity element generates the whole group.
Galois associated to each polynomial a group of symmetries of its roots. A polynomial is solvable by radicals (has a formula like the quadratic formula) if and only if its Galois group is solvable. Since the symmetric group S₅ is not solvable, the general degree-5 polynomial has no radical formula.
Lagrange (1771) observed permutation constraints on polynomial roots. Ruffini (1799) and Abel (1824) independently proved quintic insolubility. Galois (1830–32) introduced the term "group" and connecting group structure to solvability. The abstract axiomatic definition emerged in the 1880s–90s (Cayley, Dyck, Weber).
If φ : G → H is a group homomorphism, then G / ker(φ) ≅ im(φ). Combined with the Second and Third Isomorphism Theorems, this is the toolkit for analysing group structure.
Every finite simple group belongs to one of: cyclic groups ℤ/pℤ, alternating groups Aₙ (n ≥ 5), 16 infinite families of Lie type, or 26 sporadic groups. Proof completed 2004 — tens of thousands of pages. The largest sporadic group — the Monster group — has order ~8 × 10⁵³, connected to modular forms and string theory (Monstrous Moonshine, Borcherds, Fields Medal 1998).
Noether's theorem (1915): every continuous symmetry of a physical system corresponds to a conserved quantity. Rotational symmetry → angular momentum conservation. The Standard Model gauge group is U(1) × SU(2) × SU(3).