A periodic counterexample with locally finite involution centralizers
We construct, for every odd integer $n\geq 665$, an infinite group $G_n$ generated by three involutions, of exponent dividing $4n$, such that the centralizer of every involution is locally finite. More precisely, each such centralizer has a normal elementary abelian $2$-subgroup of finite index. Moreover, the three generators are conjugate in $G_n$, so that $G_n$ is normally generated by one involution. The construction gives a negative answer to Problem~20.56 of the Kourovka Notebook. Apart from Adian's result that free Burnside groups of sufficiently large odd exponent are infinite, all the arguments are elementary and are proved in detail.
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Lemma 3.1
Let $D$ be a group with a normal elementary abelian $2$-subgroup $E$ of finite index. Then $D$ 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--New York, 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--New York, 1979.
- External referenceD. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996.LaTeX source\bibitem{Robinson} D.~J.~S. Robinson, \emph{A Course in the Theory of Groups}, 2nd ed., Graduate Texts in Mathematics, vol.~80, Springer-Verlag, New York, 1996.
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- PF:2026.000026/v1/THM-1.1
For every odd integer $n\geq 665$ there exists a group $G_n$ with the following properties: \item $G_n$ is infinite and generated by three involutions; \item $g^{4n}=1$ for every $g\in G_n$; \item for every involution $t\in G_n$, the centralizer $\CG{G_n}{t}$ contains a normal elementary abelian $2$-subgroup of finite index; \item $G_n$ is generated by the conjugacy class of a single involution. Consequently, $G_n$ is an infinite finitely generated periodic group, all of whose involution centralizers are locally finite.
- PF:2026.000026/v1/LEM-2.1
For every $x\in\Omega$ the point stabilizer $\Stab_Q(x)$ has order two. Consequently, if $S\subseteq\Omega$ is finite and nonempty, then \[ |\Stab_Q(S)|\leq 2|S|, \] where $\Stab_Q(S)=\{q\in Q:q\cdot S=S\}$ is the setwise stabilizer.
- PF:2026.000026/v1/LEM-3.1
Let $D$ be a group with a normal elementary abelian $2$-subgroup $E$ of finite index. Then $D$ is locally finite.
- PF:2026.000026/v1/PROP-3.2
Let $H$, $Q$, $V$, and $K$ be as in Section 2, with $n$ odd. For every involution $z\in K$, the centralizer $\CG Kz$ contains a normal elementary abelian $2$-subgroup of finite index. In particular, $\CG Kz$ is locally finite.
- PF:2026.000026/v1/LEM-4.1
The elements $a$ and $b$ have order exactly $n$ in $H$. Furthermore, the assignments \[ a\longmapsto a^{-1}, \qquad b\longmapsto b^{-1} \] define an automorphism $\alpha$ of $H$ with $\alpha^2=1$.
- PF:2026.000026/v1/LEM-4.2
Each of $s_0,s_1,s_2$ is a nonidentity involution belonging to $K$. Moreover, the restriction of $\pr$ to $G_n$ is surjective onto $Q$.
- PF:2026.000026/v1/LEM-4.3
The three involutions $s_0,s_1,s_2$ are conjugate in $G_n$. Consequently, \[ G_n=\langle s_0^g:g\in G_n\rangle. \]
- PF:2026.000026/v1/COR-4.4
For every odd $n\geq665$ there exists an infinite three-generated group of exponent dividing $4n$, normally generated by an involution, in which the centralizer of every element of even order is locally finite.