Kourovka Notebook Problem 19.2

Definitions and mathematical statement

For a group $G$ and a subgroup $L\leq G$, define the core of $L$ in $G$ by \[ L_G=\bigcap_{g\in G}gLg^{-1}, \] the largest normal subgroup of $G$ contained in $L$. Write $1$ for the trivial subgroup. Following the supplied transcription, for $M\trianglelefteq G$, say that $G$ nearly splits over $M$ if there exists $N\leq G$ such that \[ [G:N]=\infty,\qquad [G:MN]<\infty,\qquad (M\cap N)_G=1. \] The proposed question is whether, for every group $G$, every prime $p$, and every normal subgroup $H\trianglelefteq G$ of order $p$, \[ \psi(G)\cap H=1 \quad\Longleftrightarrow\quad G\text{ nearly splits over }H. \] Unresolved qualifications. The supplied record does not define $\psi(G)$, so the left-hand side cannot be given a complete mathematical interpretation from the supplied evidence. Moreover, the two index requirements in the transcribed definition are incompatible when $M$ is finite, hence in particular when $|H|=p$. This is a formulation issue requiring source clarification; no correction to the definition or to the hypothesis on $H$ is assumed here.

Short English statement

M. K. Azarian asks whether a normal subgroup $H$ of prime order has trivial intersection with $\psi(G)$ exactly when $G$ nearly splits over $H$, using the supplied definition above. The notation $\psi(G)$ is undefined in the record, and the stated index conditions conflict with the finiteness of $H$. The question therefore remains an unresolved transcription requiring review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701550$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 19.2; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The record attributes the problem to M. K. Azarian, proposed in 2018, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), original problem number 19.2, together with the 21st issue edition (2026), identified as arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels this problem open. This is supplied source metadata, not an independently verified mathematical or literature assessment. The cited publication has not been consulted for this draft; no definition of $\psi(G)$ or explanation of the index conflict is supplied.