Groups of Prime Power Order Covered by a Certain Number of Proper Subgroups

Abstract

Let G be a group. A set of proper subgroups of G is called a cover or covering for G if its set-theoretic union is equal to G. A cover for G is called irredundant if every proper subset of the cover is not again a cover for G. Yakov Berkovich proposed the following problem: does there exist a p-group G admitting an irredundant covering by n subgroups, where $$p+1<n<2p$$ ? If ‘yes’, classify such groups. We prove that for any prime $$p\ge 3$$ , every finite p-group whose minimum number of generators is at least 3 has an irredundant cover of size $$\frac{3(p+1)}{2}$$ . It follows that the classification of all finite p-groups having an irredundant covering of size n where $$p+1<n<2p$$ is not possible.