Kourovka Notebook Problem 15.61

Definitions and mathematical statement

The supplied question is recorded below under the usual finite-group interpretation. The source excerpt does not explicitly specify finiteness, the permitted sets of primes, or conventions for trivial groups; these qualifications require review.

Let $\pi$ be a nonempty set of primes, and let $\pi'$ be its complement in the set of all primes. A finite $\pi$-group is a finite group whose order has prime divisors only in $\pi$. Interpret a finite group $G$ as $\pi$-soluble if it has a normal series whose factors are either $\pi'$-groups or $p$-groups for primes $p\in\pi$. A Hall $\pi$-subgroup $G_\pi\leq G$ satisfies that $|G_\pi|$ is a $\pi$-number and $[G:G_\pi]$ is a $\pi'$-number.

For a finite soluble group $H$, let $n(H)$ denote its nilpotent length: the least number of nontrivial factors in a normal series of $H$ all of whose factors are nilpotent. For a finite $p$-soluble group $G$, let $l_p(G)$ denote its ordinary $p$-length, the least number of nontrivial $p$-group factors in a normal series whose factors are either $p$-groups or $p'$-groups. Define the nilpotent $\pi$-length $l_\pi^n(G)$ to be the least number of nontrivial nilpotent $\pi$-group factors in a normal series of $G$ whose factors are either $\pi'$-groups or nilpotent $\pi$-groups. The superscript $n$ is part of the notation for this invariant, not an exponent.

Question (V. S. Monakhov; supplied source formulation). Is it true that every $\pi$-soluble group $G$, with a Hall $\pi$-subgroup $G_\pi$, satisfies \[ l_\pi^n(G)\leq n(G_\pi)-1+\max_{p\in\pi}l_p(G)? \] The finite-group definitions above give a conventional reading of this question, subject to the unresolved boundary qualifications below.

The supplied source reports an affirmative answer when every proper subgroup of $G_\pi$ is supersoluble. Here a finite group is supersoluble if it admits a normal series with cyclic factors of prime order. This affirmative special case is a supplied source claim, not an independently verified result.

Unresolved boundary qualification. With the usual empty-series conventions, $n(1)=0$ and the length invariants of the trivial group are zero. The source does not explain how its displayed inequality is intended to apply when $G_\pi=1$, nor how to interpret the maximum if $\pi=\varnothing$. No additional hypothesis or exceptional convention resolving these cases is supplied.

Short English statement

Monakhov asks whether the nilpotent $\pi$-length of a $\pi$-soluble group is at most the nilpotent length of a Hall $\pi$-subgroup, minus one, plus the largest ordinary $p$-length for $p\in\pi$. The source reports an affirmative answer when every proper subgroup of that Hall subgroup is supersoluble. Finiteness and degenerate cases require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $701149$, exact source problem code catalogue entry, title Kourovka Notebook Problem 15.61; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The supplied attribution identifies V. S. Monakhov as proposer, with proposed year 2002, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), Problem 15.61; it also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry supplies the status label open and reports the affirmative special case concerning proper supersoluble subgroups. These are supplied assessments, not independently verified mathematical or literature conclusions. No separate citation for the special case or literature-verification date is supplied.