On Some Hereditary and Super Classes of Directly Finite Abelian Groups
Peter Danchev, Brendan Goldsmith, Fatemeh Karimi
TL;DR
The paper investigates hereditary and super properties within directly finite Abelian groups by introducing Extended Bassian (EB) groups and leveraging Corner's Extension of Szele's Theorem. It establishes reduction principles: for Hopfian subclasses containing Bassian, hereditary and super classes coincide with Bassian; for directly finite subclasses containing EB, they coincide with EB. The authors classify super directly finite groups, showing they decompose as $G = B \oplus D$ with $B$ Bassian and $D = \bigoplus_p \mathbb{Z}(p^{\infty})^{(n_p)}$ (each $n_p$ finite), and prove that within $\mathcal{X}$ with $\mathcal{EB} \subseteq \mathcal{X} \subseteq \mathcal{DF}$, the star-classes satisfy $\mathcal{X}^s = \mathcal{DF}^s = \mathcal{EB}^s$, collapsing to EB. The work deepens understanding of how Bassian, EB, and direct finiteness interact in abelian group theory, with implications for structure theory and epimorphic image analysis.
Abstract
Continuing recent studies of both the hereditary and super properties of certain classes of Abelian groups, we explore in-depth what is the situation in the quite large class consisting of directly finite Abelian groups. Trying to connect some of these classes, we specifically succeeded to prove the surprising criteria that a relatively Hopfian group is hereditarily only when it is extended Bassian, as well as that, a relatively Hopfian group is super only when it is extended Bassian. In this aspect, additional relevant necessary and sufficient conditions in a slightly more general context are also proved.