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

Abstract

The coprime commutators ${\gamma }_j^{\ast }$ and ${\delta }_j^{\ast }$ 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 ${\gamma }_1^{\ast }$-commutator and a ${\delta }_0^{\ast }$-commutator. Now let $j\ge 2$ and let $X$ be the set of all elements of $G$ that are powers of ${\gamma }_{j-1}^{\ast }$-commutators. An element $g$ is a ${\gamma }_j^{\ast }$-commutator if there exist $a\in X$ and $b\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\ge 1$ let $Y$ be the set of all elements of $G$ that are powers of ${\delta }_{j-1}^{\ast }$-commutators. The element $g$ is a ${\delta }_j^{\ast }$-commutator if there exist $a,b\in Y$ such that $g=[a,b]$ and $(|a|,|b|)=1$. The subgroups of $G$ generated by all ${\gamma }_j^{\ast }$-commutators and all ${\delta }_j^{\ast }$-commutators are denoted by ${\gamma }_j^{\ast }(G)$ and ${\delta }_j^{\ast }(G)$, respectively. For every $j\ge 2$ the subgroup ${\gamma }_j^{\ast }(G)$ 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 $j\ge 1$ the subgroup ${\delta }_j^{\ast }(G)$ is precisely the last term of the lower central series of ${\delta }_{j-1}^{\ast }(G)$, that is, ${\delta }_j^{\ast }(G)={\gamma }_{\infty }({\delta }_{j-1}^{\ast }(G))$. In the present paper we prove that if $G$ possesses $m$ cyclic subgroups whose union contains all ${\gamma }_j^{\ast }$-commutators of $G$, then ${\gamma }_j^{\ast }(G)$ contains a subgroup $\Delta$, of $m$-bounded order, which is normal in $G$ and has the property that ${\gamma }_j^{\ast }(G)/\Delta$ is cyclic. If $j\ge 2$ and $G$ possesses $m$ cyclic subgroups whose union contains all ${\delta }_j^{\ast }$-commutators of $G$, then the order of ${\delta }_j^{\ast }(G)$ is $m$-bounded.

Final version, referee's suggestions added