Kourovka Notebook Problem 1.65

Definitions and mathematical statement

The supplied question is: "Is the class of groups of abelian extensions of abelian groups closed under taking direct sums $(A,B)\mapsto A\oplus B$?" The phrase "groups of abelian extensions" is not defined in the supplied record, so the intended class remains unresolved.

One possible interpretation uses extension groups. For abelian groups $X$ and $Y$, let $\operatorname{Ext}^{1}_{{\mathbb{Z}}}(X,Y)$ denote the abelian group, under Baer sum, of equivalence classes of short exact sequences of abelian groups \[ 0\longrightarrow Y\longrightarrow E\longrightarrow X\longrightarrow 0. \] Two extensions are equivalent if there is an isomorphism between their middle groups commuting with the displayed maps and inducing the identity on $X$ and $Y$. Define the class \[ \mathcal E=\{G:\ G\text{ is an abelian group and }G\cong\operatorname{Ext}^{1}_{{\mathbb{Z}}}(X,Y)\text{ for some abelian groups }X,Y\}. \] Under this interpretation, the question is whether \[ \forall A,B\in\mathcal E,\qquad A\oplus B\in\mathcal E. \] Equivalently, for every four abelian groups $X_1,Y_1,X_2,Y_2$, must there exist abelian groups $X,Y$ such that \[ \operatorname{Ext}^{1}_{{\mathbb{Z}}}(X_1,Y_1)\oplus\operatorname{Ext}^{1}_{{\mathbb{Z}}}(X_2,Y_2) \cong\operatorname{Ext}^{1}_{{\mathbb{Z}}}(X,Y)? \] Here $A\oplus B$ is the direct sum of the two abelian groups.

A different interpretation concerns groups that are themselves extensions with abelian kernel and abelian quotient. Define \[ \mathcal M=\{G:\ \text{there exists }N\trianglelefteq G\text{ such that }N\text{ and }G/N\text{ are abelian}\}. \] Under that interpretation the question would instead be \[ \forall A,B\in\mathcal M,\qquad A\oplus B\in\mathcal M, \] where the binary external direct sum means the direct product $A\times B$ with componentwise group operation. These interpretations define different classes. The supplied wording does not establish which is intended; neither formulation is asserted here to be the definitive source problem. No finiteness or countability restrictions are supplied.

Short English statement

The source asks whether a certain class associated with abelian extensions is closed under binary direct sums. Its terminology leaves unresolved whether its members are Baer extension groups $\operatorname{Ext}^{1}_{{\mathbb{Z}}}(X,Y)$ or groups having an abelian normal subgroup and abelian quotient. Both conditional formulations are recorded for review.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $700056$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 1.65." The registry labels the problem open; this is a supplied status, not an independently verified mathematical assessment.
• Supplied attribution: L. Ya. Kulikov, proposed in 1965; The Kourovka Notebook (Unsolved Problems in Group Theory), first issue (1965), Problem 1.65. The record identifies the 21st issue edition (2026), arXiv:1401.0300v48. Its collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Registry provenance: upstream ID $700056$, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82; record created and updated at 2026-10-16T14:00:00Z. These are supplied registry timestamps, not dates of independent source or literature verification.