# free group

free group
A group (set with a particular kind of binary operation) that has a presentation without relators; equivalently, a free product of some number of copies of ℤ.

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• Free group — 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 =… …   Wikipedia

• Free — may refer to: Free will Political freedom Economic freedom Something given or supplied without payment (gratis) Gratis versus Libre, the distinction between the two meanings above Free may also refer to: Contents 1 Arts and philosophy …   Wikipedia

• Free probability — is a mathematical theory which studies non commutative random variables. The freeness property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around… …   Wikipedia

• Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… …   Wikipedia

• torsion-free group — /tawr sheuhn free /, Math. a group in which every element other than the identity has infinite order. * * * …   Universalium

• torsion-free group — /tawr sheuhn free /, Math. a group in which every element other than the identity has infinite order …   Useful english dictionary

• Free abelian group — In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients …   Wikipedia

• Free product — In abstract algebra, the free product of groups constructs a group from two or more given ones. Given, for example, groups G and H , the free product G*H can be constructed as follows: given presentations of G and of H , take the generators of G… …   Wikipedia

• Free object — In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a clean… …   Wikipedia

• Group ring — This page discusses the algebraic group ring of a discrete group; for the case of a topological group see group algebra, and for a general group see Group Hopf algebra. In algebra, a group ring is a free module and at the same time a ring,… …   Wikipedia