A note on minimal coverings of groups by subgroups (Q2765547)

A cover for a group \(G\) is a finite set of subgroups whose set theoretic union is \(G\). A minimal cover is a cover (obviously irredundant) of minimal cardinality.NEWLINENEWLINENEWLINEIn their first main result, the authors continue the work of \textit{M. J. Tomkinson} [Math. Scand. 81, No. 2, 191-198 (1997; Zbl 0905.20014)] on minimal covers of finite soluble groups and prove that every minimal cover of a finite soluble noncyclic group contains at most one nonmaximal subgroup.NEWLINENEWLINENEWLINETwo other results are proved for infinite groups. The first characterizes the groups admitting a minimal cover consisting of Abelian subgroups in terms of the structure of the central factor group, while the second gives a characterization of the soluble nonnilpotent groups having a minimal cover consisting of nilpotent subgroups.