In mathematics, the free group F S over a given set S consists of all words that can be built from members of S , considering two words to be different unless their equality follows from the group axioms (e.g. st = suu −1 t but s ≠ t −1 for s ,t ,u ∈ S). The members of S are called generators of F S , and the number of generators is the rank of the free group. 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 exactly one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu −1 t).

A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property.