noun
Meaning 1
A group that has a presentation without relators; equivalently, a free product of some number of copies of Z.
Definition source: English Wiktionary via Wiktextract
Topics: group-theory, mathematics, sciences
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.
- 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.
Meaning relationships
Synonyms: none provided
Antonyms: none provided