Given a set S of "free generators" of a free group, let 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 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 or where . Noting that r(r(w)) = r(w) for any string w, define an equivalence relation such that if and only if . Then let the underlying set of the free group generated by S be the quotient set 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 groupF 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 then all take the form , where 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.