Bounding the degree of generic sharp transitivity

2024/07/12 by Tuna Altınel, Altınel, Tuna, Joshua Wiscons +1
Computer Science · Mathematics · #20B22 (Primary) 20F11 (Secondary) #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Rings, Modules, and Algebras #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2407.09636

openalex publication_date 2024/07/12 · openalex created_date 2024/07/17 · openalex updated_date 2026/07/28

Abstract

We show that a generically sharply t-transitive permutation group of finite Morley rank on a set of rank r satisfies t≤ r+2 provided the pointwise stabilizer of a generic (t-1)-tuple is an L-group, which holds, for example, when this stabilizer is solvable or when r≤ 5. This makes progress on the Borovik-Cherlin conjecture that every generically (r+2)-transitive permutation group of finite Morley rank on a set of rank r is of the form PGL_r+1(F) acting naturally on ℙ^r(F). Our proof is assembled from three key ingredients that are independent of the main theorem - these address actions of Alt(n) on L-groups of finite Morley rank, generically 2-transitive actions with abelian point stabilizers, and simple groups of rank 6.

Related

• Towards the recognition of PGL_n via a high degree of generic transitivity
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• Actions of Alt(n) on groups of finite Morley rank without involutions