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City Campus 
512.2 FREC

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 Description
 v, 99 pages ; 26 cm
 Notes
 "September 2018 . Volume 255 . Number 1219 (second of 7 numbers)."Contents: Background material  Expanded pure groups  Unipotent groups over Q and definable linearity  Definably affine groups  Tori in expanded pure groups  The definably linear quotients of an ACFgroup  The group DG and the main theorem for K = Q  The main theorem for K = Q  Biinterpretability and standard isomorphisms.Includes bibliographical references and index.Summary: "We analyze the abstract structure of algebraic groups over an algebraically closed field K. For K of characteristic zero and G a given connected affine algebraic Qgroup, the main theorem describes all the affine algebraic Qgroups H such that the groups H(K) and G(K) are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic Qgroups G and H, the elementary equivalence of the pure groups G(K) and H(K) implies that they are abstractly isomorphic. In the final chapter, we apply our results to characterize the connected algebraic groups all of whose abstract automorphisms are standard, when K is either Q or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited"