Finite coverings by subgroups with a given property

Abstract

Letbe a group-theoretic property. We say a group has a finite covering by-subgroups if it is the set-theoretic union of finitely many-subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups,n-abelian subgroups or 2-central subgroups.R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [G: ZC(G)] finite implies the existence of a finite covering by subgroups of nilpotency classc, i.e. ℜc-groups. However, an example of a group is given there which has a finite covering by ℜ2-groups, butZ2(G) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.