LEADER 03462nam 2200493 450 001 9910478892103321 005 20210901203040.0 010 $a1-4704-4815-7 035 $a(CKB)4100000007133848 035 $a(MiAaPQ)EBC5571101 035 $a(PPN)231946023 035 $a(Au-PeEL)EBL5571101 035 $a(OCoLC)1065073359 035 $a(EXLCZ)994100000007133848 100 $a20181125d2018 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAlgebraic Q-groups as abstract groups /$fOlivier Frecon 210 1$aProvidence, Rhode Island :$cAmerican Mathematical Society,$d2018. 215 $a1 online resource (v, 99 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vNumber 1219 311 $a1-4704-2923-3 327 $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) 608 $aElectronic books. 615 0$aAlgebra. 615 0$aFinite groups. 615 0$aIsomorphisms (Mathematics) 676 $a512.9 700 $aFrecon$b Olivier$01034248 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910478892103321 996 $aAlgebraic Q-groups as abstract groups$92453251 997 $aUNINA