Free group

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In mathematics, a group G is called free if there is a subset S of G such that any 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−1 = su−1ut−1). Apart from the existence of inverses no other relation exists between the generators of a free group.

A related but different notion is a free abelian group.

Contents

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.[1] 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.[2][3][4] Max Dehn realized the connection with topology, and obtained the first proof of the full Nielsen-Schreier Theorem.[5] Otto Schreier published an algebraic proof of this result in 1927,[6] and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology.[7] Later on in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras.

Examples

The group (Z,+) of integers is free; we can take S = {1}. A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there.

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