Free group

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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⁻¹t 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⁻¹t).

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.