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Aspects of free groups. (English) Zbl 1068.20027

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.

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
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