Primitive Permutation Groups of Finite Morley Rank

We prove a version of the O'Nan–Scott Theorem for definably primitive permutation groups of finite Morley rank. This yields questions about structures of finite Morley rank of the form ( F , +,., H ) where ( F , +,.) is an algebraically closed field and H is a central extension of a simple group with H ⩽ GL( n, F ). We obtain partial results on such groups H , and show for example that if char( F ) = 0, H is irreducible, and (in the sense for stable groups) some Borel subgroup of H is non‐abelian then H = Z ( H ). E where E ⩽ H is algebraic, that is, definable in ( F , +,.).