## Abstract

Let π be a set of finite groups. A finite group

*G*is called an π-cover if every group in π is isomorphic to a subgroup of*G*. An π-cover is called*minimal*if no proper subgroup of*G*is an π-cover, and*minimum*if its order is smallest among all π-covers. We prove several results about minimal and minimum π-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 {β€*,β€*_{q}*} has finitely many minimal covers, where*_{r}*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.Original language | English |
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Pages (from-to) | 345-372 |

Number of pages | 28 |

Journal | Journal of Algebra |

Volume | 660 |

Early online date | 23 Jul 2024 |

DOIs | |

Publication status | E-pub ahead of print - 23 Jul 2024 |

## Keywords

- Cauchy's theorem
- Cayleyβs theorem
- Simple groups
- Abelian groups