An Infinite Family of Primitive Groups of Twisted Wreath Type Attaining the Upper Bound for the Minimal Subdegree

Let \(G\) be a finite primitive permutation group of twisted wreath type, with socle \(T^k\) and primitive component \(H\). Giudici, Li, Praeger, Seress and Trofimov (2006) proved that the least non-trivial subdegree of \(G\) satisfies \(\operatorname{MinSubDeg}(G)\le k\cdot\operatorname{MinSubDeg}(H)\), and asked whether this inequality is always strict (their Question 1.6; Question 4 of Praeger and Seress in Oberwolfach Report 12/2005). An earlier note of the author gave a negative answer with the group \(\operatorname{PSL}(2,4)\operatorname{twr}\operatorname{PGL}(3,4)\). Here we show that the bound is attained by an infinite family. For a prime power \(q\ge4\) let \(P=\operatorname{PGL}(3,q)\) act on the \(k=q^2+q+1\) points of the projective plane \(\operatorname{PG}(2,q)\), let \(Q\) be the stabiliser of a point \(y\), and let \(Q\) act on \(T=\operatorname{PSL}(2,q)\) by conjugation, through its action on the quotient space \(\mathbb{F}_q^3/y\). Then \(G_q=T\operatorname{twr} P\) is primitive of twisted wreath type, and \(\operatorname{MinSubDeg}(G_q)=k\cdot\operatorname{MinSubDeg}(H)=q(q^3-1)(q+1)/c_q\), where \(c_q\) is \(q+1\), \(2(q-1)\) or \(2(q+1)\) according as \(q\) is even, \(q\equiv1\) or \(q\equiv3\pmod 4\). By the reduction of the five authors this is a statement about the subgroups of \(\operatorname{PGL}(3,q)\) of order greater than \(q^2(q-1)c_q\). We prove it for \(q\ge5\) by a written argument from the list of the maximal subgroups of \(\operatorname{SL}(3,q)\) (Mitchell; Hartley; tabulated by Bray, Holt and Roney-Dougal), Dickson's list of the subgroups of \(\operatorname{PGL}(2,q)\) and an inverse-transpose duality; for \(q=4\) it is the result of the earlier note. Exhaustive computations for small \(q\), with independently written programs, are tests and not part of the proof. No infinite family attaining the bound was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. This is an unrefereed note.