free group

In mathematics, the free group F_S over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st = suu⁻¹t, but s ≠ t⁻¹ for s,t,u∈S). The members of S are called generators of F_S. An arbitrary group G is called free if it is isomorphic to F_S for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu⁻¹t).