Towards the recognition of PGL_n via a high degree of generic transitivity

2017/10/02 by Tuna Altınel, Altınel, Tuna, Joshua Wiscons +1
Mathematics · #03C60 #20B22 #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1710.00445

openalex publication_date 2017/10/02 · openalex created_date 2017/10/20 · openalex updated_date 2026/07/28

Abstract

In 2008, Borovik and Cherlin posed the problem of showing that the degree of generic transitivity of an infinite permutation group of finite Morley rank (X,G) is at most n+2 where n is the Morley rank of X. Moreover, they conjectured that the bound is only achieved (assuming transitivity) by PGL_n+1(\mathbbF) acting naturally on projective n-space. We solve the problem under the two additional hypotheses that (1) (X,G) is 2-transitive, and (2) (X-\x\,G_x) has a definable quotient equivalent to (ℙ^n-1(\mathbbF),PGL_n(\mathbbF)). The latter hypothesis drives the construction of the underlying projective geometry and is at the heart of an inductive approach to the main problem.

Citations

• Recognizing PGL₃ via generic 4-transitivity

Related

• Bounding the degree of generic sharp transitivity
• Recognizing PGL₃ via generic 4-transitivity
• Actions of Alt(n) on groups of finite Morley rank without involutions
• Sharply 2-transitive groups of finite Morley rank
• Groups Acting Generically Multiply Transitively on Solvable Groups