Kourovka Notebook Problem 17.123

Definitions and mathematical statement

For a finite group $G$, let its element-order spectrum be \[ \pi_e(G)=\{\operatorname{ord}(g):g\in G\}. \] A composition factor of $G$ is a simple quotient occurring in a composition series of $G$. A section of a group $K$ is a quotient $H/N$, where $H\le K$ and $N\trianglelefteq H$.

The supplied statement refers to Archive problem 13.63 for definitions, but that entry is not included. In particular, the meaning of $h$ cannot be confirmed from the supplied record. To make a conditional formulation explicit, for a finite set $S\subseteq{\mathbb{Z}}_{>0}$, define \[ h_{\mathrm{fin}}(S)=\#\{[K]:K\text{ is a finite group and }\pi_e(K)=S\}, \] where $[K]$ denotes the isomorphism class of $K$.

Problem (conditional on the intended definition of $h$). Under the interpretation $h=h_{\mathrm{fin}}$, do there exist finite groups $G_1,G_2$ such that \[ \pi_e(G_1)=\pi_e(G_2),\qquad h_{\mathrm{fin}}(\pi_e(G_1))<\infty, \] and such that, for every $i,j\in\{1,2\}$ with $i\ne j$, every nonabelian composition factor $F$ of $G_i$, every subgroup $H\le G_j$, and every normal subgroup $N\trianglelefteq H$, one has \[ F\not\cong H/N? \] The identification of the source's $h$ with $h_{\mathrm{fin}}$ requires verification against Archive 13.63. The supplied statement does not separately require $G_1\not\cong G_2$ or require either group to have a nonabelian composition factor; no such hypotheses have been added.

Short English statement

Conditionally interpreting $h$ as the number of finite-group isomorphism classes with a prescribed element-order spectrum, ask whether two finite groups can share a spectrum realized by only finitely many such classes while every nonabelian composition factor of either group fails to occur as a section of the other. The referenced definition of $h$ remains to be checked.

Sources

• Local UnsolvedMath registry, permanent ID $701424$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 17.123"; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 17.123, 17th issue (2010); proposed by W. J. Shi. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The source explicitly refers to Archive problem 13.63 for definitions. Its contents were not supplied, so those definitions have not been verified.
• The registry supplies the status "open" and an update timestamp of 2026-10-16T14:00:00Z. These are source metadata, not an independently verified literature assessment.