Algebraic Q-groups as abstract groups / / Olivier Frécon

• Record Nr.: UNINA9910793296603321
• Autore: Frécon Olivier <1974->
• Pubbl/distr/stampa: Providence, Rhode Island : , : American Mathematical Society, , [2018]; ©2018
• ISBN: 1-4704-4815-7
• Descrizione fisica: 1 online resource (v, 99 pages)
• Collana: Memoirs of the American Mathematical Society ; ; Number 1219
• Disciplina: 512.9
• Soggetti: Algebra; Finite groups; Isomorphisms (Mathematics)
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: "September 2018 . Volume 255 . Number 1219 (second of 7 numbers)."
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Cover -- Title page -- Chapter 1. Introduction -- 1.1. Related work -- 1.2. The field of definition -- 1.3. Overview of the paper -- Chapter 2. Background material -- 2.1. Groups of finite Morley rank -- 2.2. Fundamental theorems -- 2.3. Decent tori and pseudo-tori -- 2.4. Unipotence -- Chapter 3. Expanded pure groups -- Chapter 4. Unipotent groups over \ov{\Q} and definable linearity -- Chapter 5. Definably affine groups -- 5.1. Definition and generalities -- 5.2. The subgroup ( ) -- 5.3. The subgroup ( ) -- Chapter 6. Tori in expanded pure groups -- Chapter 7. The definably linear quotients of an -group -- 7.1. The subgroups ( ) and ( ) -- 7.2. The nilpotence of ( ) -- 7.3. The subgroup ( ) when the ground field is \ov{\Q} -- 7.4. The subgroups ( ) and ( ) in positive characteristic -- Chapter 8. The group _{ } and the Main Theorem for =\ov{\Q} -- Chapter 9. The Main Theorem for ≠\ov{\Q} -- Chapter 10. Bi-interpretability and standard isomorphisms -- 10.1. Positive characteristic and bi-interpretability -- 10.2. Characteristic zero -- Acknowledgements -- Bibliography -- Index of notations -- Index -- Back Cover.
• Sommario/riassunto: The author analyzes the abstract structure of algebraic groups over an algebraically closed field K. For K of characteristic zero and G a given connected affine algebraic \overline{\mathbb Q}-group, the main ; theorem describes all the affine algebraic \overline{\mathbb Q} -groups 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 \overline{\mathbb Q} -groups G and H, the elementary equivalence of the pure groups G(K) and H(K) implies that they are abstractly isomorphic. In the final section, the author applies his results to characterize the connected algebraic groups, all of whose abstract automorphisms are standard, when K is either \overline {\mathbb Q} or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited.