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19.96 (2018)

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Let $M$ be a free metabelian group of finite rank $r \geqslant 2$, and let $P$ be the set of its primitive elements. (An element of a relatively free group is said to be primitive if it can be included in a basis of the group.)
$\qquad$ a) Is $P$ a first-order definable set?
$\qquad$ b) Is the set of bases of the group $M$ a first-order definable set?

Contributor: E. I. Timoshenko

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