Group partitions of minimal size

A cover of a finite group $G$ is a family of proper subgroups of $G$ whose union is $G$, and a cover is called minimal if it is a cover of minimal cardinality. A partition of $G$ is a cover such that the intersection of any two of its members is $\{1\}$. In this paper we determine all finite groups that admit a minimal cover that is also a partition. We prove that this happens if and only if $G$ is isomorphic to $C_p \times C_p$ for some prime $p$ or to a Frobenius group with Frobenius kernel being an abelian minimal normal subgroup and Frobenius complement cyclic.