Completely Inert Subgroups of Abelian Groups

TL;DR

The paper investigates completely inert and uniformly completely inert subgroups within abelian groups. It defines these notions and proves the key equivalences with their characteristically inert counterparts. It derives structural consequences, including commensurability with characteristic and fully invariant subgroups and preservation under direct sums. It also shows that completely inert subgroups can be not totally inert, and situates results within the broader context of inert subgroup theory and prior work on totally inert subgroups.

Abstract

We define and study in-depth the so-called completely inert and uniformly completely inert subgroups of Abelian groups. We curiously show that a subgroup is completely inert exactly when it is characteristically inert. Moreover, we prove that a subgroup is uniformly completely inert precisely when it is uniformly characteristically inert. These two statements somewhat strengthen recent results due to Goldsmith-Salce established for totally inert subgroups in J. Commut. Algebra (2025). Some other closely relevant things are obtained as well.

Theorems & Definitions (26)

• Definition 1.1
• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Proposition 2.7
• Lemma 2.8
• Lemma 2.9