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G. Cutolo and C. Nicotera
Verbal sets and cyclic coverings
J. Algebra, 324, (2010), pp. 1616–1624.
doi: 10.1016/j.jalgebra.2010.06.025
MathSciNet Zentralblatt Abstract Full Text
Abstract
We consider groups $G$ such that the set of all values of a fixed word $w$ in $G$ is covered by a finite set of cyclic subgroups. Fernández-Alcober and Shumyatsky studied such groups in the case when $w$ is the word $[x_1,x_2]$ and proved that in this case the corresponding verbal subgroup $G'$ is either cyclic or finite. Answering a question asked by them, we show that this is far from being the general rule. However, we prove a weaker form of their result in the case when $w$ is either a lower commutator word or a non-commutator word, showing that in the given hypothesis the verbal subgroup $w(G)$ must be finite-by-cyclic. Even this weaker conclusion is not universally valid: it fails for verbose words.
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