Minimal cover groups

Let $\mathcal{F}$ be a set of finite groups. A finite group $G$ is called an \emph{$\mathcal{F}$-cover} if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. An $\mathcal{F}$-cover is called \emph{minimal} if no proper subgroup of $G$ is an $\mathcal{F}$-cover, and \emph{minimum} if its order is smallest among all $\mathcal{F}$-covers. We prove several results about minimal and minimum $\mathcal{F}$-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 $\{\mathbb{Z}_q,\mathbb{Z}_r\}$ has finitely many minimal covers, where $q$ and $r$ are distinct primes. Motivated by this, we say that $n$ is a \emph{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.