free group

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free group

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Noun

free group (plural free groups)

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

   Given a set S of "free generators" of a free group, let ${\displaystyle S^{-1}}$ 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 ${\displaystyle (S\cup S^{-1})^{*}}$ 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 ${\displaystyle xx^{-1}}$ or ${\displaystyle x^{-1}x}$ where ${\displaystyle x\in S}$. Noting that r(r(w)) = r(w) for any string w, define an equivalence relation ${\displaystyle \sim }$ such that ${\displaystyle u\sim v}$ if and only if ${\displaystyle r(u)=r(v)}$. Then let the underlying set of the free group generated by S be the quotient set ${\displaystyle (S\cup S^{-1})^{*}/\sim }$ and let its operator be concatenation followed by reduction.

   • 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:

     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 ${\displaystyle V}$ then all take the form ${\displaystyle H/V(H)}$, where ${\displaystyle H}$ is a suitably chosen absolutely free group.

   • 2006, Anthony W. Knapp, Basic Algebra, Springer, page 303:

     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.

Usage notes

• If some generators are said to be free, then the group that they generate is implied to be free as well.
• The cardinality of the set of free generators is called the rank of the free group.

Translations

Further reading

• Free group on Encyclopedia of Mathematics
• Free Group on Wolfram MathWorld