Result: Packing and Covering Groups with Subgroups: Packing and covering groups with subgroups

The authors study the problem of covering or packing a finite group \(G\) with subgroups of a specified order \(s\). A collection of subgroups of order \(s\) is called an \(s\)-cover if the union of these subgroups contains every element of \(G\), and is called an \(s\)-packing if the subgroups are disjoint (ignoring the identity element). Improved upper and lower bounds on the size of \(p^t\)-packings and \(p^t\)-covers in Abelian \(p\)-groups are obtained. Some of the results concern groups that are Abelian but not elementary Abelian, and through these results a characterization of the elementary Abelian groups by the existence of large packings and small covers is obtained. The reviewer remarks that the problem discussed in the paper is closely related to that of constructing perfect single-error-correcting codes as shown by \textit{M. Herzog} and \textit{J. Schönheim} [Inf. Control 18, 364-368 (1971; Zbl 0229.94007)].