Kourovka Notebook Problem 11.44

Definitions and mathematical statement

For a prime $p$ and a finite $p$-group $X$, define its sectional rank by \[ r(X)=\max\left\{d\geq 0:\ H/K\cong ({\mathbb{Z}}/p{\mathbb{Z}})^d,\ K\trianglelefteq H\leq X\right\}. \] Equivalently, this is the maximum, over subgroups of $X$, of their minimum numbers of generators. In particular, $r(1)=0$.

A uniform formulation of the question attributed to L. S. Kazarin is: does there exist a real constant $C\geq 0$ such that, for every prime $p$, every finite $p$-group $G$, and every pair of subgroups $A,B\leq G$ satisfying \[ G=AB=\{ab:a\in A,\ b\in B\}, \] one has \[ r(G)\leq C\left(r(A)+r(B)\right)? \] Neither normality of the factors nor trivial intersection is assumed. The displayed bound expresses the existence of a linear bound in the two ranks using a common coefficient. The source does not explicitly specify whether the coefficients must be independent of $p$; the formulation above requires that uniformity. If dependence on $p$ is intended, the quantifiers instead become: for every prime $p$, there exists $C_p\geq 0$ satisfying the same bound for all such $G,A,B$.

Short English statement

Kazarin asks whether the sectional rank of a finite $p$-group that is a product of two subgroups is bounded linearly by their sectional ranks. The supplied record labels the problem open; the intended uniformity in the prime $p$ requires confirmation.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700736$, exact source code catalogue entry, title Kourovka Notebook Problem 11.44; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was updated on 2026-10-16 and supplies the status open; this status has not been independently verified.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), Problem 11.44, proposed by L. S. Kazarin. The background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.