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Lower Bounds for the Finite Quotients of Conder's Group on the Cubic Tree (Kourovka Notebook Problem 16.46)

Year: 2026
Type: preprint
Abstract: Problem 16.46 of the Kourovka Notebook, posed by M. Conder and going back to his paper of 1992, concerns the free product G = G₃ ∗_{G₁} G₄ of two of the Djoković–Miller groups that act arc-transitively on the cubic tree, amalgamated along the modular group G₁, and the normal subgroup K of index 8 of G, which has a presentation with six generators and 22 relators: does K have a non-trivial finite quotient? Conder found no proper subgroup of finite index. We do not decide the question. We prove, without computation, that K is perfect, that in every non-trivial image of K the generators keep their orders and the symmetric group S₄ embeds, and that K is the amalgamated free product of four copies of L = S₄ ∗_{D₈} S₄ over a common subgroup C₃ ∗ C₃, glued along its four sign automorphisms; a non-trivial finite quotient of K gives one of the subgroup J generated by two of these copies. By exhaustive computer searches we show that J, and hence K, has no non-trivial action on at most 36 points, so K has no subgroup of index between 2 and 36, and, with GAP, that no non-abelian finite simple group of order less than 10⁸ receives a non-trivial homomorphism from J. Hence every non-trivial finite quotient of K has order at least 10⁸. Jambor (2015) proved, by a machine computation, that K has no quotient PSL₂(q) or PGL(2,q); together with a search of the twelve other simple groups of order between 10⁸ and 10⁹, every non-trivial finite quotient of K has order at least 10⁹. In contrast, the amalgam of two copies differing by the twist (A,B) ↦ (A⁻¹,B⁻¹) maps onto A₆ and PSL(2,7). The bounds are computer-assisted; the question itself remains open. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-16.46 (Kourovka Notebook, Problem 16.46, M. Conder, 16th issue, 2006).
DOI: 10.5281/zenodo.23624374
Language: English
Open Access status: green
OpenAlex ID: W7223466613
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