Submission date: 17 October 2013.

Abstract:

We analyze the abstract structure of algebraic groups over an algebraically closed field K, using techniques from the theory of groups of finite Morley rank.

For K of characteristic zero and G a given connected affine algebraic (\bar Q)-group, the main theorem describes the algebraic structure of all the groups H(K) isomorphic as abstract groups to G(K), with H an affine algebraic (\bar Q)-group. A model theoretical consequence is that the elementary equivalence of the pure groups G(K) and H(K) implies the abstract isomorphy.

Along the way, we characterize the connected algebraic groups all of whose abstract automorphisms are standard, when K is either (\bar Q) or of positive characteristic.

Mathematics Subject Classification: 03C60, 14L17, 20E36, 20G15

Keywords and phrases: Algebraic groups, Groups of finite Morley rank, Abstract isomorphisms, Elementary equivalence, Burdges' unipotence.

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