|
Abstract
|
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 Fcovers. We prove several results about minimal and minimum
F-covers: for example, every minimal cover of a set of pgroups (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 {Zq, Zr}
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.
|