Kourovka Notebook Problem 21.132

Definitions and mathematical statement

For a prime $p$, an abstract $p$-group means a group in which every element has finite order equal to a power of $p$; no common exponent bound or local-finiteness assumption is imposed. A group $G$ is finitely generated if $G=\langle S\rangle$ for a finite set $S$. It is residually finite if for every $g\ne1$ there are a finite group $F$ and a homomorphism $\varphi:G\to F$ with $\varphi(g)\ne1$. Its centre is $Z(G)=\{z\in G:zg=gz\text{ for all }g\in G\}$.

The source requests, for every prime $p$, a construction based on a development of E. S. Golod's construction of a finitely generated residually finite $p$-group $G_p$ satisfying \[1<|Z(G_p)|<\infty.\] Thus the centre must be nontrivial and finite; a trivial centre or an infinite centre is not the requested case. The method restriction "based on the development of E. S. Golod's construction" is part of the source request, although the record does not define a formal class of admissible Golod constructions.

There is an important missing qualifier to check: the displayed group-theoretic conditions do not explicitly require $G_p$ itself to be infinite. Without a meaningful construction restriction or an infinitude requirement, finite cyclic groups of order $p$ already satisfy the bare listed properties. The Golod context strongly matters, but it is not a substitute for a stated hypothesis. This draft therefore does not silently add "infinite" or treat the bare finite-group observation as a solution of the intended method-specific problem.

The discussion attributes the question to A. V. Timofeenko and cites developments of Golod's construction in Discrete Mathematics and Applications 23, nos. 5--6 (2013), 491--501. It reports known comparison groups with infinite centres and with trivial centres, pointing to Kourovka 9.76 and Archive 11.101. The supplied literature triage checked 2026-08-17 says no Golod-based construction for the requested nontrivial finite-centre case was verified and labels the intended question open (OPEN-TRIAGE); the registry is open. Those dated source assessments and the unresolved construction/infinitude convention are preserved together.

Short English statement

For each prime, construct a finitely generated residually finite $p$-group with a finite nontrivial centre using the intended development of Golod's construction. The source leaves that construction class and an expected infinitude requirement unstated, so the bare finite-group existence question must be distinguished from the intended one.

Sources

• Local UnsolvedMath registry, permanent ID 2641, source code catalogue entry, "Kourovka Notebook Problem 21.132"; supplied construction request and A. V. Timofeenko attribution.
• Kourovka Notebook, 21st issue (2026), Problem 21.132, supplied maintained source: https://www.kourovka-notebook.org/.
• Development of E. S. Golod's construction, Discrete Mathematics and Applications 23, nos. 5--6 (2013), 491--501, as supplied; the record gives no author, article title, or URL for this reference.
• Kourovka 9.76 and Archive 11.101, supplied discussion cross-references for known infinite-centre and trivial-centre examples; no permanent UnsolvedMath IDs are inferred.
• Supplied literature triage checked 2026-08-17, OPEN-TRIAGE; it reports no verified requested Golod-based construction, records no source alteration, and cautions that dated triage does not exclude later or unindexed results.