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1.19 (1965)

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(A. I. Mal’cev). Which subgroups (subsets) are first order definable in a free group? Which subgroups are relatively elementarily definable in a free group? In particular, is the derived subgroup first order definable (relatively elementarily definable) in a free group?

Contributor: Yu. L. Ershov

Progress

All definable (and relatively elementarily definable) sets are described; in particular, a subgroup is relatively elementarily definable if and only if it is cyclic (O. Kharlampovich, A. Myasnikov, Int. J. Algebra Comput., 23, no. 1 (2013), 91–110).

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