Kourovka Notebook Problem 20.32

Definitions and mathematical statement

Let $G$ be a group with identity $e$, and let $A\subseteq G$ be a symmetric generating set, meaning \[ A=A^{-1}:=\{a^{-1}:a\in A\},\qquad \langle A\rangle=G. \] The literal finite-list question in the supplied record is whether, for every such pair $(G,A)$, there exist a positive integer $n$ and elements $a_1,\ldots,a_n\in A$ such that the map \[ \{1,\ldots,n\}\longrightarrow G,\qquad k\longmapsto p_k:=a_1a_2\cdots a_k \] is a bijection. Thus the partial products must be pairwise distinct and exhaust $G$; the generators $a_i$ themselves need not be distinct.

Unresolved domain qualification. The source does not state that $G$ is finite. A finite list exhausting $G$ entails finiteness and $n=|G|$. If finite groups are the intended domain, the universal quantifier above must explicitly be restricted to finite groups; this restriction is not an express hypothesis of the supplied record.

Boundary and graph qualifications. The displayed list begins at $p_1=a_1$, not at the empty product $p_0=e$. The identity must occur exactly once among $p_1,\ldots,p_n$, but the source does not require $p_n=e$ or specify a closing edge. Neither $e\notin A$ nor $A\ne\varnothing$ nor $|G|>1$ is imposed. The introductory assertion concerning the Lovász conjecture uses the word "Hamiltonian" without specifying a path or a cycle. Its intended relationship to the displayed partial-product question remains unresolved; no additional closing condition is imposed here.

Short English statement

Given a symmetric generating set of a group, can one choose a finite sequence of its generators whose successive products enumerate every group element exactly once? The source leaves finiteness unstated and does not specify a Hamiltonian path or cycle convention. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701690$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 20.32." The registry labels the problem open; this is a supplied status, not an independently verified mathematical or literature assessment.
• The supplied background attributes Problem 20.32 to The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), and the 21st issue edition (2026), identifying arXiv:1401.0300v48: https://arxiv.org/abs/1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic attribution has not been independently checked.
• The proposer fields and background name B. Green, with proposed year 2022. The statement separately begins with the attribution "(E. Rapaport Strasser)." Both attributions are retained; their respective roles are not explained by the supplied record.
• The source invokes the Lovász conjecture in its introductory motivation. No separate citation or precise path-versus-cycle formulation is supplied for that assertion.