eBook ISBN:  9781470448158 
Product Code:  MEMO/255/1219.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470448158 
Product Code:  MEMO/255/1219.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 255; 2018; 99 ppMSC: 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.

Table of Contents

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. Biinterpretability and standard isomorphisms

Acknowledgements

Index of notations


Additional Material

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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. Biinterpretability and standard isomorphisms

Acknowledgements

Index of notations