Kourovka Notebook Problem 8.15

Definitions and mathematical statement

The expressions $\widetilde{N}$-group and $\overline{\! Z}$-group denote the classes referenced in M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of the Theory of Groups, 3rd edition, Moscow, Nauka, 1982, p. 205 (Russian). The supplied record does not reproduce those definitions; their exact hypotheses and conventions therefore remain unresolved in this draft.

Subject to those referenced definitions, the question is whether \[ \text{for every group }G,\qquad \left(G\text{ is a }\widetilde{N}\text{-group}\right) \Longrightarrow \left(G\text{ is a }\overline{\! Z}\text{-group}\right). \] The record states no additional restriction on $G$. This is a conditional transcription of the intended implication, requiring the cited definitions before it becomes a self-contained mathematical statement.

Short English statement

Does every $\widetilde{N}$-group belong to the class of $\overline{\! Z}$-groups, with both terms interpreted exactly as in the cited textbook? The definitions are absent from the supplied record and require review.

Sources

• Local UnsolvedMath registry, permanent ID $700467$, source code catalogue entry, title Kourovka Notebook Problem 8.15. The supplied registry labels the problem open; this label is not an independently verified assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 8.15, 8th issue (1982), as attributed in the record to the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by Sh. S. Kemkhadze; the statement describes it as a well-known problem.
• M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of the Theory of Groups, 3rd edition, Moscow, Nauka, 1982, p. 205 (Russian), cited by the supplied statement for the definitions of the two group classes.