A Negative Answer to a Question of Praeger and Seress on Minimal Subdegrees of Primitive Groups of Twisted Wreath Type

Abstract

Let G be a finite primitive permutation group of twisted wreath type, with socle T^k and point stabiliser P, and let H be its primitive component: the group induced on one coordinate of the product decomposition Δ^k, where |Δ| = |T|. Praeger and Seress (Oberwolfach Report 12/2005, on joint work with Giudici, Li and Trofimov) state that the least non-trivial subdegree satisfies MinSubDeg(G) ≤ k · MinSubDeg(H), and they ask in their Question 4 (p. 692) whether this inequality is always strict. We give a negative answer to Question 4. Let P = PGL(3,4) act on the 21 points of the projective plane of order 4, let Q be the stabiliser of a point y, let T = PSL(2,4) ≅ A_5, and let Q act on T through its action on the quotient space F_4^3 / y. The twisted wreath product G = T twr P, acting on its base group, is primitive of twisted wreath type of degree 60^21, with k = 21, H = A_5 × A_5, MinSubDeg(H) = 12 and MinSubDeg(G) = 252 = 21 · 12. By a reduction which is known in substance, and which we prove again, the last equality is a statement about the subgroups of PGL(3,4) of order greater than 240. We prove this statement in two ways: computer-assisted, by exhaustive exact computations with independently written programs; and by a written argument from the list of the maximal subgroups of PSL(3,4) and PGL(3,4) in the tables of Connor and Leemans. Computer-assisted, we also find equality for the top groups PΓL(3,4) (315 = 21 · 15) and PSL(3,5) (465 = 31 · 15), where H is the full holomorph of A_5, and strict inequality for PSL(3,4) and PΣL(3,4). The full paper of Giudici, Li, Praeger, Seress and Trofimov (2006) and its companion paper were not accessible to us; we make no statement that depends on their content. No answer to Question 4 was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. We claim neither that the example is the smallest one nor a classification of the cases of equality, and the question remains open under additional hypotheses. 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: OWR-793-006 (Praeger–Seress, On minimal subdegrees of finite primitive permutation groups, Oberwolfach Report 12/2005, Question 4).