Packing and Covering Groups with Subgroups

We study the problem of covering or packing a finite group with subgroups of a specified order and obtain bounds on the size of such covers and packings. Our main results provide characterizations of the elementary abelian groups by the existence of large packings or small covers, respectively. Hence large packings and small covers can be thought of as geometric objects: they correspond to large partial t-spreads and small t-covers of a suitable projective space PG(d, p) for some prime p. We shall also exhibit some series of examples which show that our bounds are reasonable.