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Algebraic $\overline{\mathbb{Q}}$-Groups as Abstract Groups

Olivier Frécon Laboratoire de Mathématiques et Applications, Université de Poitiers, Poitiers, France
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Softcover ISBN: 978-1-4704-2923-2
Product Code: MEMO/255/1219
List Price: $78.00 MAA Member Price:$70.20
AMS Member Price: $46.80 Electronic ISBN: 978-1-4704-4815-8 Product Code: MEMO/255/1219.E List Price:$78.00
MAA Member Price: $70.20 AMS Member Price:$46.80
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AMS Member Price: $70.20 Click above image for expanded view Algebraic$\overline{\mathbb{Q}}$-Groups as Abstract Groups Olivier Frécon Laboratoire de Mathématiques et Applications, Université de Poitiers, Poitiers, France Available Formats:  Softcover ISBN: 978-1-4704-2923-2 Product Code: MEMO/255/1219  List Price:$78.00 MAA Member Price: $70.20 AMS Member Price:$46.80
 Electronic ISBN: 978-1-4704-4815-8 Product Code: MEMO/255/1219.E
 List Price: $78.00 MAA Member Price:$70.20 AMS Member Price: $46.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$70.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 2552018; 99 pp
MSC: Primary 20; Secondary 03; 14;

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.

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

• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 2552018; 99 pp
MSC: Primary 20; Secondary 03; 14;

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.

• Chapters
• 1. Introduction
• 2. Background material
• 3. Expanded pure groups
• 4. Unipotent groups over $\bar {\mathbb {Q}}$ and definable linearity
• 5. Definably affine groups
• 6. Tori in expanded pure groups
• 7. The definably linear quotients of an $ACF$-group
• 8. The group $D_G$ and the Main Theorem for $K=\bar {\mathbb {Q}}$
• 9. The Main Theorem for $K\neq \bar {\mathbb {Q}}$
• 10. Bi-interpretability and standard isomorphisms
• Acknowledgements
• Index of notations
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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