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

Abstract

The coprime commutators $γ_j^*$ and $δ_j^*$ 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. They are defined as follows. Let $G$ be a finite group. Every element of $G$ is both a $γ_1^*$-commutator and a $δ_0^*$-commutator. Now let $j\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $γ_{j-1}^*$-commutators. An element $g$ is a $γ_j^*$-commutator if there exist $a\in X$ and $b\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\geq 1$ let $Y$ be the set of all elements of $G$ that are powers of $δ_{j-1}^*$-commutators. The element $g$ is a $δ_j^*$-commutator if there exist $a,b\in Y$ such that $g=[a,b]$ and $(|a|,|b|)=1$. The subgroups of $G$ generated by all $γ_j^*$-commutators and all $δ_j^*$-commutators are denoted by $γ_j^*(G)$ and $δ_j^*(G)$, respectively. For every $j\geq2$ the subgroup $γ_j^*(G)$ is precisely the last term of the lower central series of $G$ (which throughout the paper is denoted by $γ_\infty(G)$) while for every $j\geq1$ the subgroup $δ_j^*(G)$ is precisely the last term of the lower central series of $δ_{j-1}^*(G)$, that is, $δ_j^*(G)=γ_\infty(δ_{j-1}^*(G))$. In the present paper we prove that if $G$ possesses $m$ cyclic subgroups whose union contains all $γ_j^*$-commutators of $G$, then $γ_j^*(G)$ contains a subgroup $Δ$, of $m$-bounded order, which is normal in $G$ and has the property that $γ_{j}^{*}(G)/Δ$ is cyclic. If $j\geq2$ and $G$ possesses $m$ cyclic subgroups whose union contains all $δ_j^*$-commutators of $G$, then the order of $δ_j^*(G)$ is $m$-bounded.