Completely realisable groups

Given a construction $f$ on groups, we say that a group $G$ is \textit{$f$-realisable} if there is a group $H$ such that $G\cong f(H)$, and \textit{completely $f$-realisable} if there is a group $H$ such that $G\cong f(H)$ and every subgroup of $G$ is isomorphic to $f(H_1)$ for some subgroup $H_1$ of $H$ and vice versa. In this paper, we determine completely ${\rm Aut}$-realisable groups. We also study $f$-realisable groups for $f=Z,F,M,D,\Phi$, where $Z(H)$, $F(H)$, $M(H)$, $D(H)$ and $\Phi(H)$ denote the center, the Fitting subgroup, the Chermak-Delgado subgroup, the derived subgroup and the Frattini subgroup of the group $H$, respectively.Comment: To appear in J. Algebra App

Relationships between Almost Completely Decomposable Abelian Groups with Their Multiplication Groups

For an Abelian group $G$, any homomorphism $\mu\colon G\otimes G\rightarrow G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of all multiplications on an Abelian group $G$ is an Abelian group with respect to addition. An Abelian group $G$ with multiplication, defined on it, is called a \textsf{ring on the group} $G$. Let $\mathcal{A}_0$ be the class of Abelian block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In the paper, we study relationships between the above groups and their multiplication groups. It is proved that groups from $\mathcal{A}_0$ are definable by their multiplication groups. For a rigid group $G\in\mathcal{A}_0$, the isomorphism problem is solved: we describe multiplications from $\text{Mult}\,G$ that define isomorphic rings on $G$. We describe Abelian groups that are realized as the multiplication group of some group in $\mathcal{A}_0$. We also describe groups in $\mathcal{A}_0$ that are isomorphic to their multiplication groups.Comment: arXiv admin note: text overlap with arXiv:2205.1065

Balanced Butler Groups

AbstractThe class of Butler groups, pure subgroups of finite rank completely decomposable groups, has been studied extensively by abelian group theorists in recent years. Classification by numerical invariants up to quasi-isomorphism and even isomorphism has been achieved for special subclasses. Here we highlight a new class in which to extend and expand classification results, the balanced Butler groups or K(1)-groups. These are the pure balanced subgroups of finite rank completely decomposable groups. A strictly decreasing chain of classes of Butler groups, introduced by Kravchenko, is obtained by defining the K(n)-groups (n≥2) to be those balanced subgroups of a completely decomposable group for which the quotient is a K(n−1)-group. We establish an internal characterization of K(n)-groups, give a method for constructing examples, and derive decomposition results

Almost completely decomposable groups and unbounded representation type

AbstractAlmost completely decomposable groups with a regulating regulator and a p-primary regulator quotient are studied. It is shown that there are indecomposable such groups of arbitrarily large rank provided that the critical typeset contains some basic configuration and the exponent of the regulator quotient is sufficiently large