## 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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### References:

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