free group

Noun

 * 1)  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.
 * 1) * 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 free groups in $$V$$ then all take the form $$H/V(H)$$, where $$H$$ is a suitably chosen absolutely free group.

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: frjáls grúpa
 * Italian: gruppo libero