Aspects of free groups.
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.
scientific article; zbMATH DE number 1924684