Kourovka Notebook Problem 10.51

Definitions and mathematical statement

Let $p$ be a prime. An abelian $p$-group is an abelian group $G$, written additively, such that \[ \forall x\in G\;\exists n\in{\mathbb{Z}}_{\geq 0}\quad p^n x=0. \] Requested structural description (S. V. Rychkov, 1986). Describe the structure of thin abelian $p$-groups.

Unresolved terminology and scope. The supplied record does not define the technical adjective "thin." Consequently, it does not determine a membership condition for the class whose structure is requested. The statement must retain "thin" in its intended source-specific sense until that definition is supplied; it must not be read as applying to all abelian $p$-groups. The record also does not specify the form of the requested structural description or impose an explicit finiteness or cardinality restriction.

Supplied qualification. The source states that such a description is known within the class of separable $p$-groups, citing C. Megibben, Michigan Mathematical Journal 13, no. 2 (1966), 153--160. Thus the requested description concerns thin abelian $p$-groups in general, with the separable case reported as already understood. The record supplies neither a definition of "separable" nor the precise result established by that reference.

Short English statement

Rychkov asks for a structural description of thin abelian $p$-groups. The supplied source reports that the separable case is known. A complete mathematical formulation requires the intended definition of "thin" and clarification of the cited separable-case result.

Sources

• Local UnsolvedMath registry: permanent ID $700666$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 10.51." Attribution: S. V. Rychkov; proposed in 1986. The supplied registry labels the problem open; that label is supplied curation, not an independently verified assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 10.51, 10th issue (1986). The supplied record identifies the consulted edition as the 21st issue edition (2026), arXiv:1401.0300v48. The registry describes the collection as edited by E. I. Khukhro and V. D. Mazurov. These source details have not been independently checked.
• C. Megibben, Mich. Math. J. 13, no. 2 (1966), 153--160. Supplied as the reference for the reported structural description in the separable case. No article title or precise theorem statement is supplied.