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.
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.
Lemma 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 $\alpha^2=1$.
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--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.
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 · 8 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:2026.000026/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.