LEADER 03640nam 2200529 450 001 9910793296603321 005 20220525075748.0 010 $a1-4704-4815-7 035 $a(CKB)4100000007133848 035 $a(MiAaPQ)EBC5571101 035 $a(Au-PeEL)EBL5571101 035 $a(OCoLC)1065073359 035 $a(RPAM)20662073 035 $a(PPN)231946023 035 $a(EXLCZ)994100000007133848 100 $a20220525d2018 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAlgebraic Q-groups as abstract groups /$fOlivier Fre?con 210 1$aProvidence, Rhode Island :$cAmerican Mathematical Society,$d[2018] 210 4$dİ2018 215 $a1 online resource (v, 99 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vNumber 1219 300 $a"September 2018 . Volume 255 . Number 1219 (second of 7 numbers)." 311 $a1-4704-2923-3 320 $aIncludes bibliographical references and index. 327 $aCover -- 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. 330 $aThe author analyzes the abstract structure of algebraic groups over an algebraically closed field K. For K of characteristic zero and G a given connected affine algebraic \overline{\mathbb Q}-group, the main theorem describes all the affine algebraic \overline{\mathbb Q} -groups H such that the groups H(K) and G(K) are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic \overline{\mathbb Q} -groups G and H, the elementary equivalence of the pure groups G(K) and H(K) implies that they are abstractly isomorphic. In the final section, the author applies his results to characterize the connected algebraic groups, all of whose abstract automorphisms are standard, when K is either \overline {\mathbb Q} or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited. 410 0$aMemoirs of the American Mathematical Society ;$vNumber 1219. 606 $aAlgebra 606 $aFinite groups 606 $aIsomorphisms (Mathematics) 615 0$aAlgebra. 615 0$aFinite groups. 615 0$aIsomorphisms (Mathematics) 676 $a512.9 700 $aFre?con$b Olivier$f1974-$01544050 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910793296603321 996 $aAlgebraic Q-groups as abstract groups$93797936 997 $aUNINA