Babai Polylogarithmic Diameter Conjecture for Finite Simple Groups

Definitions and mathematical statement

Let $G$ be a nontrivial finite simple group, meaning that its only normal subgroups are $\{1_G\}$ and $G$. A subset $A\subseteq G$ generates $G$ if $\langle A\rangle=G$. For a positive integer $\ell$, define the exact product set \[ A^\ell=\{a_1\cdots a_\ell:a_1,\ldots,a_\ell\in A\}. \] The supplied assertion is that there exists an absolute real constant $C>0$, independent of $G$ and $A$, such that \[ \forall\,G\text{ finite simple}\quad \forall\,A\subseteq G\text{ with }\langle A\rangle=G\quad \exists\,\ell\in\mathbb Z_{\geq1}: \quad \ell\leq(\log|G|)^C\quad\text{and}\quad A^\ell=G. \] Unresolved formulation qualifications. The source does not assume $1_G\in A$ or $A=A^{-1}$, does not exclude cyclic simple groups, and does not specify the logarithm base. With the exact-product definition above, a singleton generating set in a cyclic group has singleton powers, so the displayed assertion cannot hold over the stated domain. This is a defect in the supplied formulation requiring clarification, rather than an accepted formulation of the intended conjecture.

For comparison of conventions only, put $A^0=\{1_G\}$. Products of at most $L$ elements of $A$ form $\bigcup_{j=0}^{L}A^j$, whereas the usual undirected word ball uses $\bigcup_{j=0}^{L}(A\cup A^{-1})^j$. These differ from the source's exact product set. The record does not establish which convention or additional hypotheses were intended; none is substituted here. The missing logarithm convention also affects the literal bound for small groups because no multiplicative constant or small-group exception is stated.

Short English statement

The supplied target asks for one absolute exponent giving a polylogarithmic product-length bound for every generating subset of every finite simple group. Its exact-length wording and inclusion of cyclic simple groups create an obstruction. The intended group domain, word-length convention, and logarithm normalization require clarification before this can serve as a precise statement of Babai's conjecture.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006243$, exact source problem code catalogue entry, title Babai Polylogarithmic Diameter Conjecture for Finite Simple Groups; collection: Oberwolfach Reports Open Problems; proposed year: 2025. No separate proposer is specified.
• Supplied report citation: Finite Groups, Fusion Systems and Applications (2025), Oberwolfach Reports, pp. 689--752, DOI 10.4171/owr/2025/16. https://doi.org/10.4171/owr/2025/16. This is also the supplied literature-source link labelled "current workshop statement." No exact page for this problem is supplied.
• The supplied excerpt attributes the workshop contribution On The Liebeck--Nikolov--Shalev Conjecture to Noam Lifshitz, joint work with Lásló Pyber, Endre Szabó, and Nick Gill; names are retained as supplied. It identifies the target as Babai's conjecture. Its contextual reference to a stronger Liebeck--Nikolov variant for sets that are not necessarily normal does not provide a separate complete formulation or its hypotheses.
• Supplied literature source: "Biswas, large-rank diameter bounds." https://doi.org/10.1112/jlms.12018. The record supplies no further bibliographic details or precise theorem statement.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the conjecture open and reports known bounded-rank and large-rank subcases and results for special generating sets. This is supplied dated curation, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, with the qualification that only formula delimiters, notation, grammar, and typography were repaired and that mathematical content and scope were preserved. That label does not establish mathematical correctness or resolve the formulation issues above.