Kourovka Notebook Problem 13.67

Definitions and mathematical statement

An involution in a group $G$ is an element $i$ of order exactly $2$. Write $i^g=g^{-1}ig$ and $C_G(i)=\{x\in G:xi=ix\}$. A group is periodic if every element has finite order. A group $H$ is residually periodic if, for every $h\in H\setminus\{1\}$, there exists a normal subgroup $N\trianglelefteq H$ such that $h\notin N$ and $H/N$ is periodic; explicitly, \[ \forall x\in H\;\exists n\in{\mathbb{Z}}_{\geq 1}\quad (xN)^n=N. \] The supplied source refers to the definition of a $T_0$-group in Kourovka Notebook Problem 12.101, but does not supply that definition. Thus the hypothesis that $G$ is a $T_0$-group remains unresolved here, and the question below is conditional on the intended meaning of that term.

For every $T_0$-group $G$ and every involution $i\in G$ such that \[ G=\left\langle i^g\mid g\in G\right\rangle, \] must $C_G(i)$ be residually periodic? Equivalently, under these hypotheses, must every $h\in C_G(i)\setminus\{1\}$ survive in some periodic quotient of $C_G(i)$?

Short English statement

V. P. Shunkov asks whether the centralizer of an involution whose conjugates generate a $T_0$-group is residually periodic. The source's definition of a $T_0$-group is missing, so the intended class of groups requires clarification from Problem 12.101.

Sources

• Local UnsolvedMath registry, permanent ID $700985$, exact source code catalogue entry, title "Kourovka Notebook Problem 13.67"; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: V. P. Shunkov, proposed in 1995; The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), Problem 13.67. The supplied record identifies the 21st issue edition (2026), arXiv:1401.0300v48. This bibliographic attribution is supplied source data and has not been independently verified.
• The statement refers to Kourovka Notebook Problem 12.101 for the definition of a $T_0$-group; that referenced content is absent from the supplied record.
• The registry labels this problem "open". This is a supplied status label, not an independently verified assessment of the literature.