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Finitely Generated Groups and First‐Order Logic

It is proved that the following classes of finitely generated groups have Π11 ‐complete first‐order theories: all finitely generated groups, the n‐generated groups, and the strictly n‐generated groups (n⩾2). Moreover, all those theories are distinct. Similar techniques show that quasi‐finitely axiomatizable groups have a hyperarithmetical word problem, where a finitely generated group is quasi‐finitely axiomatizable if it is the only finitely generated group satisfying an appropriate first‐order sentence. The Turing degrees of word problems of quasi‐finitely axiomatizable groups form a cofinal set in the Turing degrees of hyperarithmetical sets.