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.