Nies, André Aspects of free groups. (English) Zbl 1068.20027 J. Algebra 263, No. 1, 119-125 (2003). Let \(F_k\) be a free group of rank \(k\). Let \(\overline g\) and \(\overline h\) be \(n\)-tuples of \(F_2\) which satisfy the same existential formulas. It is proved here that there exists an automorphism of \(F_2\) which maps \(\overline g\) to \(\overline h\). The following results are typical: (1) Fix a prime \(p\). \(F_k\) is, up to isomorphism, the only \(k\)-generated group having all finite \(k\)-generated \(p\)-groups as homomorphic images. (2) The theory of non-Abelian free groups has no prime model. Another result worth mentioning is that if the endomorphism \(\alpha\colon F_k\to F_k\) preserves \(\exists\)-formulas in both directions, then \(\alpha\) is an automorphism. – The reference list contains 12 articles. Reviewer: Stylianos Andreadakis (Athens) Cited in 1 ReviewCited in 8 Documents MSC: 20E05 Free nonabelian groups 20A15 Applications of logic to group theory 20E36 Automorphisms of infinite groups 03C60 Model-theoretic algebra 03D25 Recursively (computably) enumerable sets and degrees Keywords:existential formulas; free groups; model theory; automorphisms; endomorphisms PDF BibTeX XML Cite \textit{A. Nies}, J. Algebra 263, No. 1, 119--125 (2003; Zbl 1068.20027) Full Text: DOI OpenURL References: [1] Baumslag, G.; Myasnikov, A.; Shpilrain, V., Open problems in combinatorial group theory, (), 1-27, also see Magnus homepage at · Zbl 0976.20023 [2] Bergman, G., Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups, Trans. amer. math. soc., 351, 1531-1550, (1999) · Zbl 0926.20016 [3] Hodges, W., Model theory, Encyclopedia math., (1993), Cambridge Univ. Press Cambridge [4] Kargapolov, M.; Merzljakov, J., Fundamentals of the theory of groups, (1979), Springer-Verlag · Zbl 0549.20001 [5] Kharlampovich, O.; Myasnikov, A., Tarski’s problem about the elementary theory of free groups has a positive solution, Electron. res. announc. amer. math. soc., 4, 101-108, (1998) · Zbl 0923.20016 [6] Lyndon, R.C.; Schupp, P.E., Combinatorial group theory, (1977) · Zbl 0368.20023 [7] Magnus, W., Über freie faktorgruppen und freie untergruppen gegebener gruppen, Monatsh. math. phys., 47, 307-313, (1939) · JFM 65.0059.03 [8] Neumann, H., Varieties of groups, Ergeb. math. grenzgeb., 37, (1967) · Zbl 0149.26704 [9] A. Nies, Separating classes of groups by first-order formulas, Internat J. Algebra Comput., to appear · Zbl 1059.20002 [10] Pickel, F., A property of finitely generated residually finite groups, Bull. amer. math. soc., 15, 347-350, (1976) · Zbl 0354.20024 [11] Robinson, D., A course in the theory of groups, (1988), Springer-Verlag [12] Turner, E., Test words for automorphisms of free groups, Bull. London math. soc., 28, 255-263, (1996) · Zbl 0852.20022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.