A group is a closed club with identity and undo buttons
Take a set and a way to combine any two members into a third still in the set. Demand associativity, an identity element, and an inverse for every element. That short rulebook—illustrated by integers under addition—organizes symmetries across mathematics and physics.
The integers with addition are the everyday model: adding is associative, zero is the identity, and negatives undo. Formally a group pairs a set with a binary operation that obeys those three axioms; uniqueness of identity and inverses follows rather than being assumed. Notation splits into additive groups, with identity often written 0, and multiplicative groups, where juxtaposition hides the operation symbol and the identity is often 1. If the operation also commutes, the group is abelian; nonabelian groups usually stick to multiplicative notation.
The idea unifies numbers, geometric shapes, and polynomial roots. Symmetry groups of objects, transformation groups of a given type, Lie groups in geometry and the Standard Model, the Poincaré group of spacetime symmetries, and point groups in molecular chemistry all fit the same pattern. Évariste Galois coined the French term groupe in the 1830s for symmetries of equation roots. After input from number theory and geometry the abstract notion solidified around 1870. Modern group theory studies groups for their own sake—via subgroups, quotients, simple groups, representations, and computation.
Finite group theory reached a landmark when the finite simple groups were fully classified, a project finished in 2004. From the mid-1980s onward, geometric group theory has treated finitely generated groups as geometric objects. Function composition, permutation multiplication, and matrix multiplication supply concrete operations when elements are maps rather than numbers. Wherever there is a reversible symmetry you can compose, there is likely a group quietly counting the ways.
Source: Group (mathematics)
