On finite groups in which coprime commutators are covered by few cyclic subgroups

The coprime commutators γj∗ and δj∗ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. Every element of a finite group G is both a γ1∗-commutator and a δ0∗-commutator. Now let j>=2 and let X be the set of all elements of G that are powers of γj-1∗ -commutators. An element g is a γj∗ -commutator if there exist a ∈ X and b ∈ G such that g = [a,b] and (|a|,|b|) = 1. For j>=1 let Y be the set of all elements of G that are powers of δj-1∗ -commutators. An element g is a δj∗ -commutator if there exist a,b ∈ Y such that g = [a,b] and (|a|,|b|) = 1. The subgroups of G generated by all γj∗-commutators and all δj∗-commutators are denoted by γj∗(G) and δj∗(G), respectively. For every j >=2 the subgroup γj∗(G) is precisely the last term γ∞(G) of the lower central series of G, while for every j>=1 the subgroup δj∗(G) is precisely the last term of the lower central series of δj∗−1(G), that is, δj∗(G) = γ∞ (δj-1∗ (G)). In the present paper we prove that if G possesses m cyclic subgroups whose union contains all γj∗-commutators of G, then γj∗(G) contains a subgroup Δ, of m-bounded order, which is normal in G and has the property that γj∗(G)/Δ is cyclic. If j>=2 and G possesses m cyclic subgroups whose union contains all δj∗-commutators of G, then the order of δj∗(G) is m-bounded.