Kourovka Notebook Problem 12.57

Definitions and mathematical statement

Let $G$ be a finite group and let $A\leq G$ be cyclic of order $4$. For $g\in G$, write $A^g=g^{-1}Ag$. Say that $A$ is a $TI$-subgroup if \[ \forall g\in G,\qquad A\cap A^g\in\{\{1\},A\}. \] In particular, no assumption that $A$ is nonnormal is imposed. A component of $G$ is a subnormal quasisimple subgroup $L\leq G$: thus there is a chain $L=L_0\trianglelefteq L_1\trianglelefteq\cdots\trianglelefteq L_r=G$, and $L=[L,L]$ with $L/Z(L)$ a nonabelian simple group. Here $Z(L)$ denotes the center of $L$. Write \[ C_G(L)=\{g\in G: g\ell=\ell g\text{ for every }\ell\in L\}. \] The source notation $C(L)$ is interpreted as $C_G(L)$; the ambient centralizer is not explicitly defined in the supplied record. The product $LC_G(L)$ is a subgroup of $G$.

1. (a) Is it true that, for every such pair $(G,A)$ and every component $L$ of $G$, \[ A\cap LC_G(L)=\{1\}\quad\Longrightarrow\quad A\leq C_G(L)? \] No additional hypothesis that $A$ normalizes $L$ is supplied.
2. (b) Assuming that $A\leq C_G(L)$ for every component $L$ of $G$, determine the structure of the normal closure \[ \langle A^G\rangle=\langle A^g:g\in G\rangle. \] The assumption includes groups with no components, for which it is vacuous. The source does not specify the desired form of the structural description or classification.

Short English statement

For a cyclic $TI$-subgroup $A$ of order $4$ in a finite group $G$, ask whether trivial intersection with $LC_G(L)$ forces $A$ to centralize a component $L$. Also determine the structure of the normal closure of $A$ when $A$ centralizes every component of $G$.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700876$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.57. The registry labels the problem open; this is a supplied status, not an independently verified mathematical or literature assessment.
• Attributed in the supplied record to A. A. Makhnëv, proposed in $1992$, as Problem 12.57 of The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992). The record also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied attribution and have not been independently verified.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of Kourovka Notebook (Issues 1--21), covering 1965--2026. No separate literature-verification date or additional literature-source links are supplied.