Kourovka Notebook Problem 21.98

Definitions and mathematical statement

Use the commutator convention $[a,b]=a^{-1}b^{-1}ab$. A multilinear commutator word, also called an outer commutator word, is formed recursively from distinct variables by bracketing words that involve disjoint sets of variables. For example $[[x_1,x_2],[x_3,x_4]]$ is such a word; no variable is repeated. Regard $w=w(x_1,\ldots,x_r)$ as a fixed word in a free group.

For an arbitrary group $G$, its set of $w$-values and its verbal subgroup are \[ G_w=\{w(g_1,\ldots,g_r):g_1,\ldots,g_r\in G\}, \qquad w(G)=\langle G_w\rangle. \] Suppose there are finitely many cyclic subgroups $C_1,\ldots,C_t\le G$, with $t\ge1$, such that $G_w\subseteq C_1\cup\cdots\cup C_t$. A cyclic subgroup means one generated by a single element, possibly of finite order.

Must $w(G)$ be finite-by-cyclic? Precisely, must there be a finite normal subgroup $N\trianglelefteq w(G)$ such that $w(G)/N$ is cyclic, allowing a finite or trivial cyclic quotient? The covering assumption concerns individual values of the word. Arbitrary products of values need not lie in one of the covering cyclic subgroups. The problem asks for the stated structural conclusion for every multilinear commutator word and every group satisfying that covering assumption, without assuming finite generation of $G$.

Short English statement

If finitely many cyclic subgroups cover every value of a multilinear commutator word, must its verbal subgroup have a finite normal subgroup with cyclic quotient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2607 (KOU-21.98), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.98. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.