Kourovka Notebook Problem 12.73

Definitions and mathematical statement

A formation of finite groups is a possibly empty class of finite groups, closed under isomorphisms, homomorphic images, and subdirect products of finitely many groups with at least one factor. A subgroup of a direct product is subdirect if every coordinate projection is surjective. A finite group is soluble if its derived series terminates at the trivial group. A soluble formation is a formation all of whose members are soluble.

A formation $\mathfrak A$ is a maximal proper subformation of $\mathfrak B$ if $\mathfrak A\subsetneq\mathfrak B$ and there is no formation $\mathfrak K$ satisfying $\mathfrak A\subsetneq\mathfrak K\subsetneq\mathfrak B$. Following the supplied definition, a formation $\mathfrak H$ has length $t$ if there exists a chain \[ \varnothing=\mathfrak H_0\subsetneq\mathfrak H_1\subsetneq\cdots\subsetneq\mathfrak H_t=\mathfrak H \] in which $\mathfrak H_{i-1}$ is a maximal proper subformation of $\mathfrak H_i$ for every $1\leq i\leq t$. This is an existence condition on a chain; no uniqueness of its length is assumed. Adopt the zero-step convention for $\mathfrak H=\varnothing$, a boundary case not explicitly discussed in the source.

Let $\mathcal L_{\leq4}$ be the collection of soluble formations admitting such a chain for some integer $0\leq t\leq4$, ordered by inclusion. The source asks whether the lattice of these formations is distributive. Interpreted as a question about this ordered collection, the question is whether it is a lattice whose meet and join satisfy, for all $\mathfrak A,\mathfrak B,\mathfrak C\in\mathcal L_{\leq4}$, \[ \begin{aligned} \mathfrak A\wedge(\mathfrak B\vee\mathfrak C) &=(\mathfrak A\wedge\mathfrak B)\vee(\mathfrak A\wedge\mathfrak C),\\ \mathfrak A\vee(\mathfrak B\wedge\mathfrak C) &=(\mathfrak A\vee\mathfrak B)\wedge(\mathfrak A\vee\mathfrak C). \end{aligned} \] Here meet and join mean greatest lower bound and least upper bound in the stated ordered collection. The source calls this collection a lattice without explaining its operations or their existence. In the ambient lattice of all soluble formations, meet is intersection and join is the least formation containing both operands; whether those operations are intended to restrict to $\mathcal L_{\leq4}$ requires clarification.

Short English statement

A. N. Skiba asks whether soluble formations obtainable from the empty formation by at most four successive maximal-subformation inclusions form a distributive lattice. The source leaves the lattice operations implicit and defines length through the existence of a chain.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700891$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.73. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record labels the problem open; this is supplied source metadata, not an independently verified status assessment.
• Supplied bibliographic reference: The Kourovka Notebook (Unsolved Problems in Group Theory), original Problem 12.73 in the 12th issue (1992), as recorded in the 21st issue edition (2026), arXiv:1401.0300v48, https://arxiv.org/abs/1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. The cited edition has not been independently consulted here.
• Attribution supplied in the record: proposed by A. N. Skiba in 1992.