A bounded-exponent counterexample to Kourovka Problem 21.96
We answer Problem~21.96 of the Kourovka Notebook in the negative. If $H$ is a group without involutions, the restricted regular wreath product $\mathrm{C}_2\wr H$ has locally finite centralizers of all elements of finite even order. Moreover, this wreath product is periodic, locally finite, or finitely generated precisely when $H$ has the corresponding property. Taking $H$ to be an infinite two-generated free Burnside group of sufficiently large odd exponent yields a three-generated periodic counterexample of bounded exponent.
Evidence-backed creditSubstantive mathematical contributions
Show proposalsConcrete proposals are tied to an immutable base version. When the paper owner confirms a contribution, it can be linked to the later manuscript version that incorporated it, was revised because of it, or resolved the issue independently. If the owner has not decided within 90 days, an administrator or a Managing Editor of the paper's subject area can take an editorial decision. Proposals by co-authors are recorded but give no contribution credit.
Substantive mathematical contributions
Show proposalsConcrete proposals are tied to an immutable base version. When the paper owner confirms a contribution, it can be linked to the later manuscript version that incorporated it, was revised because of it, or resolved the issue independently. If the owner has not decided within 90 days, an administrator or a Managing Editor of the paper's subject area can take an editorial decision. Proposals by co-authors are recorded but give no contribution credit.
Theorem 2.2: Main theorem
Let $H$ be a group without involutions and let $W=\CC\wr H$ be its restricted regular wreath product. Then the centralizer of every element of finite even order in $W$ is locally finite. Furthermore, \[ W\text{ is periodic} &\iff H\text{ is periodic},\\ W\text{ is locally finite} &\iff H\text{ is locally finite},\\ W\text{ is finitely generated} &\iff H\text{ is finitely generated}. \] If $H$ has finite odd exponent $n$, then $W$ has exponent exactly $2n$.
Open the parent paper · This indexed statement is a discovery surface and does not itself imply human verification.
Mathematical dependencies
Click a node to select the same theorem, lemma or proposition in the review sidebar. The sidebar stays available while you inspect the graph. Solid arrows are explicit LaTeX references; dashed arrows are legacy structural suggestions and are never treated as proof evidence.
Versions
Every version is immutable. Open an earlier version or compare revisions here without leaving the paper reader layout.
References
ProofForum parses references embedded in the uploaded LaTeX. Internal PF IDs are linked and colored by the current human-review state of the cited paper: green = fully certified tracked structures, yellow = partial review/concern, red = incorrect point reported, grey = no human verification or unresolved internal ID. arXiv and Hexagon identifiers cited with a prefix or link are matched to ProofForum verification records: a state is shown only for the exact cited version; a citation without a version is marked as a matching work.
- External referenceS. I. Adian, The Burnside Problem and Identities in Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 95, Springer-Verlag, Berlin, 1979.LaTeX source\bibitem{Adian} S.~I.~Adian, \emph{The Burnside Problem and Identities in Groups}, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol.~95, Springer-Verlag, Berlin, 1979.
- External referenceN. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), no. 3, 385--388.LaTeX source\bibitem{GuptaSidki} N.~Gupta and S.~Sidki, \emph{On the Burnside problem for periodic groups}, Math. Z. \textbf{182} (1983), no.~3, 385--388.
- External referenceD. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 62, Springer-Verlag, Berlin, 1972.LaTeX source\bibitem{Robinson} D.~J.~S.~Robinson, \emph{Finiteness Conditions and Generalized Soluble Groups}, Part~1, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol.~62, Springer-Verlag, Berlin, 1972.
- External referenceE. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory. The Kourovka Notebook, No. 21, Sobolev Institute of Mathematics, Novosibirsk, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21tkt-1.pdf.LaTeX source\bibitem{Kourovka} E.~I.~Khukhro and V.~D.~Mazurov (eds.), \emph{Unsolved Problems in Group Theory. The Kourovka Notebook}, No.~21, Sobolev Institute of Mathematics, Novosibirsk, 2026, \url{https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21tkt-1.pdf}.
Cited by
Published ProofForum papers that cite this paper or an exact persistent object in this immutable version.
Lean / Palomar
Machine-checked Lean evidence is linked through immutable Palomar registry records. ProofForum separately tracks statement-to-declaration correspondence and independent human alignment review.
Machine-readable theorem index · 3 statements
Public text index for scholarly discovery. It mirrors theorem, lemma, proposition and related statements from this immutable version and does not imply human verification.
- PF:2026.000025/v1/LEM-2.1
If $1\ne v\in V$, $S=\supp(v)$ and $K=\Stab_H(S)$, then \[ \Cent_W(v)=K\ltimes V. \] Moreover $K$ is finite, $|K|$ divides $|S|$, and $\Cent_W(v)$ is locally finite.
- PF:2026.000025/v1/THM-2.2
Let $H$ be a group without involutions and let $W=\CC\wr H$ be its restricted regular wreath product. Then the centralizer of every element of finite even order in $W$ is locally finite. Furthermore, \[ W\text{ is periodic} &\iff H\text{ is periodic},\\ W\text{ is locally finite} &\iff H\text{ is locally finite},\\ W\text{ is finitely generated} &\iff H\text{ is finitely generated}. \] If $H$ has finite odd exponent $n$, then $W$ has exponent exactly $2n$.
- PF:2026.000025/v1/COR-3.1
For every odd integer $n\geq665$ there exists a three-generated infinite periodic group $G$ of exponent exactly $2n$ which is not locally finite, contains involutions, and satisfies \[ \Cent_G(x)\text{ is locally finite for every element $x$ of even order}. \] Consequently, Problem 21.96 of the Kourovka Notebook has a negative answer.