Algebraic \(\overline{\mathbb Q}\)-groups as abstract groups. (English) Zbl 1496.20001
Memoirs of the American Mathematical Society 1219. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2923-2/print; 978-1-4704-4815-8/ebook). v, 104 p. (2018).
Publisher’s description: 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 \(\overline{\mathbb Q}\)-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 chapter, we apply our 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.
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 chapter, we apply our 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.
MSC:
| 20-02 | Research exposition (monographs, survey articles) pertaining to group theory |
| 20F11 | Groups of finite Morley rank |
| 03C60 | Model-theoretic algebra |
| 14L17 | Affine algebraic groups, hyperalgebra constructions |
| 20E36 | Automorphisms of infinite groups |
| 20G15 | Linear algebraic groups over arbitrary fields |
Keywords:
algebraic groups; groups of finite Morley rank; abstract isomorphisms; elementary equivalence; Burdges’ unipotenceReferences:
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