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Let $w$ be a multilinear commutator word, and assume that $G$ is a group where the set of $w$-values is covered by finitely many cyclic subgroups. Is it true that the verbal subgroup $w(G)$ is finite-by-cyclic?

This is true for lower central words (G. Cutolo, C. Nicotera, J. Algebra, 324, no. 7 (2010), 1616–1624).

Contributor: M. Morigi

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