Abelian p-groups with minimal characteristic inertia

Abstract

For Abelian p-groups, Goldsmith, Salce, et al., introduced the notion of minimal full inertia. In parallel to this, we define the concept of minimal characteristic inertia and explore those p-primary Abelian groups having minimal characteristic inertia. We establish the surprising result that, for each Abelian p-group A, the square $A\oplus A$ has the minimal characteristic inertia if, and only if, it has the minimal full inertia. We also obtain some other relationships between these two properties. Specifically, we exhibit groups which do not have neither of the properties, as well as we show via a concrete complicated construction by using some nonstandard tricks from ring theory and module theory that, for any prime p, there is a p-group possessing the minimal characteristic inertia which does not possess the minimal full inertia.

Keywords:

• Abelian p-groups
• fully inert and characteristically inert subgroups
• fully transitive and transitive groups
• minimal characteristic inertia
• minimal full inertia

2020 Mathematics Subject Classification:

• 20K10

Additional information

Funding

The work of the first-named author, P.V. Danchev, is partially supported by the Bulgarian National Science Fund under grant KP-06 No. 32/1 of December 07, 2019 as well as by the Junta de Andalucía, FQM 264.