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

Abstract

The coprime commutators $\gamma_j^*$ and $\delta_j^*$ 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 $\gamma_1^*$-commutator and a $\delta_0^*$-commutator. Now let $j\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $\gamma_{j-1}^*$-commutators. An element $g$ is a $\gamma_j^*$-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 $\delta_{j-1}^*$-commutators. The element $g$ is a $\delta_j^*$-commutator if there exist $a,b\in Y$ such that $g=[a,b]$ and $(|a|,|b|)=1$. The subgroups of $G$ generated by all $\gamma_j^*$-commutators and all $\delta_j^*$-commutators are denoted by $\gamma_j^*(G)$ and $\delta_j^*(G)$, respectively. For every $j\geq2$ the subgroup $\gamma_j^*(G)$ is precisely the last term of the lower central series of $G$ (which throughout the paper is denoted by $\gamma_\infty(G)$) while for every $j\geq1$ the subgroup $\delta_j^*(G)$ is precisely the last term of the lower central series of $\delta_{j-1}^*(G)$, that is, $\delta_j^*(G)=\gamma_\infty(\delta_{j-1}^*(G))$. In the present paper we prove that if $G$ possesses $m$ cyclic subgroups whose union contains all $\gamma_j^*$-commutators of $G$, then $\gamma_j^*(G)$ contains a subgroup $\Delta$, of $m$-bounded order, which is normal in $G$ and has the property that $\gamma_{j}^{*}(G)/\Delta$ is cyclic. If $j\geq2$ and $G$ possesses $m$ cyclic subgroups whose union contains all $\delta_j^*$-commutators of $G$, then the order of $\delta_j^*(G)$ is $m$-bounded.