The element that leaves every other unchanged when combined with it
The identity element — often written e or 1, or 0 for an additive group
Science · Mathematics
Group theory
1830s – present
Mathematics — abstract algebra
The study of symmetry in the abstract. A group is a set with an operation satisfying four axioms: the operation is closed and associative, there is an identity element that leaves everything unchanged, and every element has an inverse. The idea grew out of Évariste Galois's work in the 1830s on when a polynomial equation can be solved by radicals, and it now underlies crystallography, particle physics and the Rubik's cube. Its greatest single achievement is the classification of finite simple groups, a proof spread over thousands of journal pages and completed in the 1980s.
How to spot them
- ▸Four axioms: closure, associativity, an identity element, and inverses
- ▸The mathematics of symmetry — rotations, reflections and permutations
- ▸Born from Galois's question of which equations are soluble by radicals
- ▸Abelian means the operation is commutative, after Niels Henrik Abel
- ▸The classification of finite simple groups, with its 26 sporadic exceptions
Key facts
Galois's 1832 night-before-the-duel letter on when equations are soluble by radicals (1832)
Évariste Galois, killed aged 20; the theory that bears his name is the origin of the subject
Abel's 1824 proof that the general quintic has no formula in radicals (1824)
The Abel–Ruffini theorem; 'abelian' commemorates him in lower case
The largest of the 26 sporadic simple objects, with about 8 × 10^53 elements (1982)
The Monster, constructed by Robert Griess; its 'moonshine' links to modular functions won Borcherds a Fields Medal
The theorem that the order of a subgroup divides the order of the whole
Lagrange's theorem, the first result every student meets
A cube of 43 quintillion configurations, all reachable in at most 20 moves (1974)
The Rubik's cube: its configurations form a group, and 20 is 'God's number'
The 230 ways of arranging symmetry in a three-dimensional crystal lattice
The crystallographic space groups, catalogued by Fedorov and Schoenflies in the 1890s
The 1963 theorem that odd order implies solubility, filling a whole journal issue (1963)
The Feit–Thompson theorem, whose proof filled an entire journal issue
Key ideas
- The four axioms are closure, associativity, identity and inverse. Commutativity is not required — a group with it is called abelian, and the symmetric group on three or more letters shows that most groups are not.
- Two groups that look different can be 'the same' — isomorphic — if their multiplication tables match after relabelling. Classifying groups means listing them up to isomorphism.
- Every finite group is built from finite simple groups, much as every integer is built from primes; the classification lists all the simple ones.
Quick-fire facts
- •The four group axioms are closure, associativity, the existence of an identity element and the existence of inverses
- •A group whose operation is commutative is called abelian, after Niels Henrik Abel (1802–1829)
- •Évariste Galois (1811–1832) coined the term 'group' and died at twenty after a duel, having written out his ideas the night before
- •Lagrange's theorem states that the order of any subgroup divides the order of the group
- •The classification of finite simple groups was announced complete around 1981–83 and occupies tens of thousands of pages by over a hundred authors
- •There are 18 infinite families of finite simple groups and 26 sporadic ones
- •The largest sporadic group, the Monster, has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements
- •'Monstrous moonshine', the unexpected link between the Monster and modular functions, was conjectured by Conway and Norton and proved by Richard Borcherds, who won the Fields Medal in 1998
- •Emmy Noether's theorem of 1918 ties every continuous symmetry of a physical system to a conservation law
- •Lie groups, named after Sophus Lie, describe continuous symmetries; SU(3) × SU(2) × U(1) is the symmetry group of the Standard Model
- •The symmetric group S(n), of all permutations of n objects, has n! elements; Cayley's theorem says every finite group is a subgroup of one
Drill it — 9 cards
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Sources
- MacTutor History of Mathematics Archive, University of St Andrews — The development of group theory
- Marcus du Sautoy — Symmetry: A Journey into the Patterns of Nature