Kourovka Notebook Problem 12.101

Definitions and mathematical statement

An involution in a group is an element of order exactly $2$. Write $x^g=g^{-1}xg$, $i^G=\{i^g:g\in G\}$, and $o(x)$ for the order of an element $x$. For a subgroup $H\leq G$, put \[ N_G(H)=\{g\in G:g^{-1}Hg=H\}. \] A subgroup $H$ is $i$-invariant if $H^i=H$. A $2$-subgroup is understood here to mean a subgroup every element of which has finite order a power of $2$; the source does not explicitly restrict these subgroups to be finite. A simple group is a nontrivial group whose only normal subgroups are itself and the identity subgroup.

Unresolved terminology. The supplied record does not define the periodic part of a group. Write $P(H)$ for that source term, without identifying it with either the set of finite-order elements or a particular periodic subgroup. These interpretations require confirmation, since the finite-order elements of an arbitrary group need not form a subgroup. Likewise, let $\mathcal Q$ denote the source's intended class of generalized quaternion groups. The usual finite generalized quaternion groups have presentations \[ Q_{2^n}=\langle a,b\mid a^{2^{n-1}}=1,\ b^2=a^{2^{n-2}},\ b^{-1}ab=a^{-1}\rangle, \qquad n\geq3. \] The record does not specify whether an infinite generalized quaternion group is also permitted. Finally, let $\mathcal F$ denote the source's class of Frobenius groups with abelian kernel and finite complement of even order. Its referenced definition in Problem 6.55 is not supplied; in particular, the intended convention for possibly infinite Frobenius groups remains unresolved.

Source formulation, conditional on those terminology conventions. A group $G$, together with a distinguished involution $i\in G$, satisfies the stated $T_0$ conditions when, on putting \[ C=C_G(i)=\{g\in G:gi=ig\}, \] the following hold:

1. For every $u,v\in i^G$, one has $o(uv)<\infty$.
2. Every $2$-subgroup $S\leq G$ is cyclic or belongs to $\mathcal Q$.
3. The subgroup $C$ is infinite, $C\ne G$, and its periodic part is finite: $|P(C)|<\infty$.
4. For every subgroup $H\leq G$ satisfying $1<|H|<\infty$ and $H^i=H$, \[ N_G(H)\leq C \quad\text{or}\quad P\left(N_G(H)\right)\in\mathcal F. \] In the second alternative, the periodic part is asserted to be a group with the specified Frobenius structure.
5. For every $c\in G\setminus C$ such that $o(ci)=2$, there exists $s\in C$ such that \[ \left|\langle c,c^s\rangle\right|=\infty. \]

The question is whether there exist a simple group $G$ and an involution $i\in G$ satisfying all five conditions. The distinguished involution must be the same throughout. This is a statement-only transcription with explicitly unresolved terminology, rather than a fully disambiguated definition of $T_0$.

Short English statement

Does there exist a simple group containing an involution for which all five stated $T_0$ conditions hold? These conditions constrain products of conjugate involutions, $2$-subgroups, the involution's centralizer, certain finite-subgroup normalizers, and subgroups generated by an element and one of its centralizer conjugates. The meanings of periodic part, generalized quaternion in the infinite setting, and the referenced Frobenius-group convention require confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $700917$, source code catalogue entry, title Kourovka Notebook Problem 12.101. The supplied record labels the problem open; that label is supplied registry metadata, not an independently verified literature assessment.
• V. P. Shunkov, proposer, 1992. Attributed source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), Problem 12.101; supplied edition reference: 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The statement refers to Kourovka Notebook Problem 6.55 for Frobenius groups. That referenced text is not included in the supplied record.