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We prove that if $G$ is a group of finite Morley rank which acts definably and generically sharply $n$-transitively on a connected abelian group $V$ of Morley rank $n$ with no involutions, then there is an algebraically closed field $F$ of characteristic $\ne 2$ such that $V$ has a structure of a vector space of dimension $n$ over $F$ and $G$ acts on $V$ as the group $\operatorname{GL}_n(F)$ in its natural action on $F^n$. This is the final pre-publication version of the paper: A. Berkman and A. Borovik, Groups of finite Morley rank with a generically sharply multiply transitive action, J. Algebra (2018), https://doi.org/10.1016/j.jalgebra.2018.07.033. Accepted for publication 28 July 2018. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published, Comment: in its final form

In this work, we give two characterisations of the general linear group over an algebraically closed field as a group G of finite Morley rank acting on an abelian connected group V of finite Morley rank definably (in the sense that G ⋉ V is a group of finite Morley rank in which G and V are definable), faithfully and irreducibly. We prove that if the pseudoreflection rank of G is equal to the Morley rank of V , then V has a definable vector space structure over an algebraically closed field, G ≅ GL ( V ) and the action is the natural action. The same result holds also under the assumption of Prufer 2-rank of G being equal to the Morley rank of V .

The paper contains a construction of a definable BN-pair in a simple group of finite Morley rank and even type with a sufficiently good system of 2-local parabolic subgroups. This provides ‘the final identification theorem’ for simple groups of finite Morley rank and even type.

This paper gives a partial answer to the Cherlin–Zil'ber Conjecture, which states that every infinite simple group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field. The classification of the generic case of tame groups of odd type follows from the main result of this work, which is an analogue of Aschbacher's Classical Involution Theorem for finite simple groups.

This paper provides a method for identifying “sufficiently rich” simple groups of finite Morley rank with simple algebraic groups over algebraically closed fields. Special attention is given to the even type case, and the paper contains a number of structural results about simple groups of finite Morley rank and even type.