symmetric group

The symmetric group is the group of every possible way to rearrange a set of objects. Imagine n labelled chairs and n people: a symmetric group element is one complete seating plan, and combining two elements means doing one rearrangement and then another. It collects all permutations of n things into a single structure, written Sₙ.

Formally, Sₙ is the set of all bijections from a set of n elements to itself, with the operation being function composition. Its size is n factorial, n! = n · (n-1) · … · 2 · 1, because there are n choices for where the first object goes, then n-1 for the next, and so on. Composition is associative, the identity permutation does nothing, and every rearrangement can be reversed — so the group axioms all hold.

Symmetric groups are central because of Cayley's theorem: every finite group is a subgroup of some Sₙ, so understanding symmetric groups means understanding all finite groups in disguise. For n at least 3, Sₙ is non-abelian — the order in which you perform two rearrangements genuinely matters.