Almost completely decomposable groups with a critical typeset of type $(1,3)$ and a $p$-primary regulator quotient are studied. It is shown that there are, depending on the exponent of the regulator quotient $p^k$, either no indecomposables if $k\le 2$; only six near isomorphism types of indecomposables if $k=3$; and indecomposables of arbitrary large rank if $k\ge 4$.

We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle \cap H=1$. We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.

If $H$ is an isotype knice subgroup of a global Warfield group $G$, we introduce the notion of a $k$-subgroup to obtain various necessary and sufficient conditions on the quotient group $G/H$ in order for $H$ itself to be a global Warfield group. Our main theorem is that $H$ is a global Warfield group if and only if $G/H$ possesses an $H\left({\aleph }_0\right)$-family of almost strongly separable $k$-subgroups. By an $H\left({\aleph }_0\right)$-family we mean an Axiom 3 family in the strong sense of P. Hill. As a corollary to the main theorem, we are able to characterize...

In this paper, we initiate the study of various classes of isotype subgroups of global mixed groups. Our goal is to advance the theory of $\Sigma$-isotype subgroups to a level comparable to its status in the simpler contexts of torsion-free and $p$-local mixed groups. Given the history of those theories, one anticipates that definitive results are to be found only when attention is restricted to global $k$-groups, the prototype being global groups with decomposition bases. A large portion of this paper is...