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).

A related but different notion is a free abelian group, both notions are particular instances of a free object from universal algebra.

History

Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on the hyperbolic plane). In an 1882 paper, Walther von Dyck pointed out that these groups have the simplest possible presentations. The algebraic study of free groups was initiated by Jakob Nielsen in 1924, who gave them their name and established many of their basic properties.Max Dehn realized the connection with topology, and obtained the first proof of the full Nielsen–Schreier theorem.Otto Schreier published an algebraic proof of this result in 1927, and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology. Later on in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras.