Minimal cover groups

Let 𝓕 be a set of finite groups. A finite group G is called an 𝓕-cover if every group in 𝓕 is isomorphic to a subgroup of G. An 𝓕-cover is called minimal if no proper subgroup of G is an 𝓕-cover, and minimum if its order is smallest among all 𝓕-covers. We prove several results about minimal and minimum 𝓕-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {ℤ_q,ℤ_r} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.