Co-Minimal Abelian Groups

• Autores: Shalom Feigelstock, Manfred Dugas
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 31, Nº 3, 2005, págs. 637-648
• Idioma: inglés
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• Resumen

  • Abstract. An abelian group A was called minimal in [3], if A is isomorphic to all its subgroups of finite index. We study the dual notion and call A cominimal if A is isomorphic to A/K for all finite subgroups K of A. We will see that minimal and co-minimal groups exhibit a similar behavior in some cases, but there are several di erences. While a reduced p-group A is minimal if and only if A/p!A is minimal, this no longer holds for cominimal p-groups. We show that a separable p-group A is co-minimal if and only if A is minimal. This does not hold for p-groups with elements of infinite height. We find necessary conditions for co-minimal p-groups in terms of their Ulm-Kaplansky invariants, which are also sucient for totally projective p-groups. If A is a mixed group with a knice system, also known as Axiom 3 modules, then A is co-minimal if and only if t(A), the torsion part of A, is co-minimal. We construct an example of a mixed group A such that t(A) is a totally projective p-group of length ! + 1 such that t(A) is co-minimal but A is not co-minimal. Moreover, we construct p-groups G of length ! +1 such that all Ulm-Kaplansky invariants of G are infinite, i.e. G is minimal, but G is not co-minimal.