英和和英

名詞

free group (複数形 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, 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.