Minimal cover groups (Q6612140)

Let \(\mathcal F\) be a set of finite groups. A finite group \(G\) is called an \(\mathcal F\)-cover if every group in \(\mathcal F\) is isomorphic to a subgroup of \(G\). An \(\mathcal F\)-cover is called minimal if no proper subgroup of \(G\) is an \(\mathcal F\)-cover, and minimum if its order is smallest among all \(\mathcal F\)-covers. When \(\mathcal F\) consists of all the groups of order \(n\), then the authors refer to an \(\mathcal F\)-cover as an \(n\)-cover. The paper contains several results about minimal and minimum \(\mathcal F\)-covers. Every minimal cover of a set of \(p\)-groups is a \(p\)-group (and it is interesting to notice that there are only finitely many minimal \(p^2\)-covers, but infinitely many minimal 8-covers); 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. If \(q<r\) are primes, the set \(\{\mathbb Z_q, \mathbb Z_r\}\) has only finitely many minimal covers if and only if \(q = 2\) and \(r\) is a Fermat prime. A positive natural number \(n\) is said to be a Cauchy number if there are only finitely many groups which are minimal with respect to having order divisible by \(n\). The authors prove that \(n\) is a Cauchy number if and only if one of the following holds: \(n\) is a prime power; \(n = 6\); \(n = 2p^a\), where \(p > 3\) is a Fermat prime and \(a \geq 1\).