Dictionary

free group en · NOUN

Meanings

1. A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ.
   • The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. 2006, Anthony W. Knapp, Basic Algebra, Springer, page 303:
   • 2002, Gilbert Baumslag, B.9 Free and Relatively Free Groups, Alexander V. Mikhalev, Günter F. Pilz, The Concise Handbook of Algebra, Kluwer Academic, page 102, The free groups in V then all take the form H/V(H), where H is a suitably chosen absolutely free group.
   • A subset A of a free group F is called "separable" when there is a non-trivial free factorization F = F₁ * F₂ such that each element of A is conjugate to an element of F₁ or of F₂. 1999, John R. Stallings, “Whitehead graphs on handlebodies”, in John Cossey, Charles F. Miller, Michael Shapiro, Walter D. Neumann, editors, Geometric Group Theory Down Under: Proceedings of a Special Year in Geometric Group Theory, Walter de Gruyter, page 317:
   • Given a set S of "free generators" of a free group, let S⁻¹ be the set of inverses of the generators, which are in one-to-one correspondence with the generators (the two sets are disjoint), then let (S∪S⁻¹)^* be the Kleene closure of the union of those two sets. For any string w in the Kleene closure let r(w) be its reduced form, obtained by cutting out any occurrences of the form xx⁻¹ or x⁻¹x where x∈S. Noting that r(r(w)) = r(w) for any string w, define an equivalence relation ∼ such that u∼v if and only if r(u)=r(v). Then let the underlying set of the free group generated by S be the quotient set (S∪S⁻¹)^*/∼ and let its operator be concatenation followed by reduction.

Forms

Translations (2)

is frjáls grúpa (group whose presentation consists of generators) · it gruppo libero (group whose presentation consists of generators)