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Algebraic $\mathbb Q$-groups as abstract groups


About this Title

Olivier Frécon

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 255, Number 1219
ISBNs: 978-1-4704-2923-2 (print); 978-1-4704-4815-8 (online)
DOI: https://doi.org/10.1090/memo/1219
Published electronically: June 21, 2018
Keywords:Algebraic groups, groups of finite Morley rank, abstract isomorphisms, elementary equivalence, Burdges’ unipotence.

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Background material
  • Chapter 3. Expanded pure groups
  • Chapter 4. Unipotent groups over $\ov \Q $ and definable linearity
  • Chapter 5. Definably affine groups
  • Chapter 6. Tori in expanded pure groups
  • Chapter 7. The definably linear quotients of an $ACF$-group
  • Chapter 8. The group $D_G$ and the Main Theorem for $K=\ov \Q $
  • Chapter 9. The Main Theorem for $K\neq \ov \Q $
  • Chapter 10. Bi-interpretability and standard isomorphisms
  • Acknowledgements
  • Index of notations

Abstract


We analyze the abstract structure of algebraic groups over an algebraically closed field .For of characteristic zero and a given connected affine algebraic -group, the main theorem describes all the affine algebraic -groups such that the groups and are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic -groups and , the elementary equivalence of the pure groups and 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 is either or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited.

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