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.
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Lemma 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.
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- 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}.
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- 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.