Kourovka Notebook Problem 11.51

Definitions and mathematical statement

All groups below are abelian and written additively. A group $G$ is torsion-free if $ng=0$, with $g\in G$ and an integer $n\geq 1$, implies $g=0$. For groups $C,A$, let $\operatorname{Ext}_{{\mathbb{Z}}}^{1}(C,A)$ denote the abelian group of equivalence classes of extensions \[ 0\longrightarrow A\longrightarrow E\longrightarrow C\longrightarrow 0, \] with its Baer sum operation.

The question proposed by A. Mader asks whether there exist "large, non-trivial" classes $\mathfrak X$ of torsion-free abelian groups such that \[ \forall A,C\in\mathfrak X,\qquad \operatorname{Ext}_{{\mathbb{Z}}}^{1}(C,A)\text{ is torsion-free}. \] Equivalently, the required extension-group condition is \[ \forall A,C\in\mathfrak X\ \forall n\in{\mathbb{Z}}_{\geq 1}\ \forall \xi\in\operatorname{Ext}_{{\mathbb{Z}}}^{1}(C,A), \qquad n\xi=0\ \Longrightarrow\ \xi=0. \] The source does not specify what "large" or "non-trivial" means for the class $\mathfrak X$. Consequently, the displayed condition is precise, but the intended existence problem remains qualified by an unspecified requirement on admissible classes. No restriction on rank, cardinality, or finite generation is supplied.

The record additionally reports, attributing the assertion to E. L. Lady and A. Mader, that two groups belonging to such a class are nearly isomorphic if and only if they have equal $p$-ranks for every prime $p$. This is retained as an attributed background assertion. The supplied record does not define near isomorphism or $p$-rank, or state any further hypotheses needed for that assertion; its precise interpretation and scope require review.

Short English statement

Find classes of torsion-free abelian groups for which $\operatorname{Ext}_{{\mathbb{Z}}}^{1}(C,A)$ is torsion-free for every ordered pair of members. The intended classes must be "large, non-trivial", but those requirements are undefined in the supplied source. The accompanying assertion about near isomorphism and $p$-ranks is attributed background whose definitions and scope require clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700743$, exact source problem code catalogue entry, title Kourovka Notebook Problem 11.51; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is supplied source metadata, not an independently verified status assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 11.51, proposed by A. Mader, 11th issue (1990). The supplied record identifies its reference edition as the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• E. L. Lady and A. Mader, J. Algebra 140 (1991), 36--64, cited in the supplied statement for the assertion relating near isomorphism to equality of all $p$-ranks. No article title is supplied.