Abelian p-groups with minimal characteristic inertia

TL;DR

This paper introduces the concept of minimal characteristic inertia for Abelian p-groups, compares it with minimal full inertia, and finds conditions under which these properties coincide or differ, including explicit constructions.

Contribution

It defines minimal characteristic inertia for Abelian p-groups and explores its relationship with minimal full inertia, providing new examples and theoretical insights.

Findings

A   has minimal characteristic inertia if and only if it has minimal full inertia.

Existence of p-groups with minimal characteristic inertia but not minimal full inertia.

Explicit constructions demonstrating differences between the two properties.

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 from ring/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.