Kourovka Notebook Problem 6.28

Definitions and mathematical statement

Let $G$ be a finite group and let $A\leq G$ be an elementary abelian $2$-group, meaning a finite-dimensional vector space over ${\mathbb{F}}_2$ under addition. Write $A^g=g^{-1}Ag$ for $g\in G$. Interpret the assertion that $A$ is a trivial-intersection ($TI$) subgroup as \[ \forall g\in G,\qquad A^g=A\quad\text{or}\quad A\cap A^g=\{1\}. \] Let $P$ be a Sylow $2$-subgroup of $G$ containing $A$. Using the conventional interpretation of weak closure, put \[ W_G(A,P)=\langle A^g\mid g\in G,\ A^g\leq P\rangle. \] The additional hypothesis is that, for some such $P$, this subgroup is abelian: \[ \forall x,y\in W_G(A,P),\qquad xy=yx. \] Structural investigation requested by the source. Describe the structure of the finite groups $G$ admitting an elementary abelian $TI$-subgroup $A$ and a Sylow $2$-subgroup $P\geq A$ satisfying these conditions.

The source asks to investigate structure; it does not specify a conjectured conclusion, classification equivalence, or required form of description. The definitions of $TI$ and weak closure above make their conventional meanings explicit, but are not supplied definitions. The record gives no rank bound for $A$, no explicit nontriviality or properness requirement, and no hypothesis that conjugates of $A$ generate $G$. Whether additional conventions or hypotheses are intended requires review.

Short English statement

Investigate finite groups containing an elementary abelian $2$-subgroup whose distinct conjugates intersect trivially and whose weak closure in a containing Sylow $2$-subgroup is abelian. The source specifies a structural investigation without a proposed conclusion; its intended scope and subgroup conventions require mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $700374$, exact source code catalogue entry, title Kourovka Notebook Problem 6.28. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied registry labels the problem open; this label has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 6.28, sixth issue (1978); supplied reference to the twenty-first issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This reference is supplied attribution and has not been independently consulted.
• Proposed by A. A. Makhnëv, with proposed year 1978, according to the supplied record.