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Elementary theory of free non-abelian groups. (English) Zbl 1110.03020
For a group \(G,\) the elementary theory \(\text{Th}(G)\) of \(G\) is the set of all first-order sentences in the language of group theory which are true in \(G.\) Around 1945 Tarski formulated two conjectures about the elementary theory of a free group. The first of them states that the elementary theory of non-abelian free groups of different ranks coincide; the second one states that the elementary theory of free groups is decidable. The scope of this paper is to prove these two conjectures.

MSC:
03C65 Models of other mathematical theories
20E05 Free nonabelian groups
20A15 Applications of logic to group theory
03B25 Decidability of theories and sets of sentences
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