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Recognizing \(\mathrm{PGL}_3\) via generic 4-transitivity. (English) Zbl 1494.20005

Summary: We show that the only transitive and generically 4-transitive action of a group of finite Morley rank on a set of Morley rank 2 is the natural action of \(\mathrm{PGL}_3\) on the projective plane.

MSC:

20B22 Multiply transitive infinite groups
03C60 Model-theoretic algebra
20F11 Groups of finite Morley rank
20G40 Linear algebraic groups over finite fields
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