research

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

Abstract

The coprime commutators γj\gamma_j^* and δj\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 GG be a finite group. Every element of GG is both a γ1\gamma_1^*-commutator and a δ0\delta_0^*-commutator. Now let j2j\geq 2 and let XX be the set of all elements of GG that are powers of γj1\gamma_{j-1}^*-commutators. An element gg is a γj\gamma_j^*-commutator if there exist aXa\in X and bGb\in G such that g=[a,b]g=[a,b] and (a,b)=1(|a|,|b|)=1. For j1j\geq 1 let YY be the set of all elements of GG that are powers of δj1\delta_{j-1}^*-commutators. The element gg is a δj\delta_j^*-commutator if there exist a,bYa,b\in Y such that g=[a,b]g=[a,b] and (a,b)=1(|a|,|b|)=1. The subgroups of GG generated by all γj\gamma_j^*-commutators and all δj\delta_j^*-commutators are denoted by γj(G)\gamma_j^*(G) and δj(G)\delta_j^*(G), respectively. For every j2j\geq2 the subgroup γj(G)\gamma_j^*(G) is precisely the last term of the lower central series of GG (which throughout the paper is denoted by γ(G)\gamma_\infty(G)) while for every j1j\geq1 the subgroup δj(G)\delta_j^*(G) is precisely the last term of the lower central series of δj1(G)\delta_{j-1}^*(G), that is, δj(G)=γ(δj1(G))\delta_j^*(G)=\gamma_\infty(\delta_{j-1}^*(G)). In the present paper we prove that if GG possesses mm cyclic subgroups whose union contains all γj\gamma_j^*-commutators of GG, then γj(G)\gamma_j^*(G) contains a subgroup Δ\Delta, of mm-bounded order, which is normal in GG and has the property that γj(G)/Δ\gamma_{j}^{*}(G)/\Delta is cyclic. If j2j\geq2 and GG possesses mm cyclic subgroups whose union contains all δj\delta_j^*-commutators of GG, then the order of δj(G)\delta_j^*(G) is mm-bounded.Comment: Final version, referee's suggestions adde

    Similar works