Free group

English

Noun

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

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   • 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.

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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

• Icelandic: Template:T
• Italian: Template:T

Further reading

• Free group on Template:W
• Free Group on Template:W