Meaning of free group | Babel Free

Definitions

A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ.

Equivalents

Examples

“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.”

“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.”

“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.”

“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.”

CEFR level