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$A_1$ Subgroups of Exceptional Algebraic Groups
 
R. Lawther Lancaster University, Lancaster, England
D. M. Testerman University of Warwick, Coventry, England
Front Cover for A_1 Subgroups of Exceptional Algebraic Groups
Available Formats:
Electronic ISBN: 978-1-4704-0265-5
Product Code: MEMO/141/674.E
List Price: $51.00
MAA Member Price: $45.90
AMS Member Price: $30.60
Front Cover for A_1 Subgroups of Exceptional Algebraic Groups
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  • Front Cover for A_1 Subgroups of Exceptional Algebraic Groups
  • Back Cover for A_1 Subgroups of Exceptional Algebraic Groups
$A_1$ Subgroups of Exceptional Algebraic Groups
R. Lawther Lancaster University, Lancaster, England
D. M. Testerman University of Warwick, Coventry, England
Available Formats:
Electronic ISBN:  978-1-4704-0265-5
Product Code:  MEMO/141/674.E
List Price: $51.00
MAA Member Price: $45.90
AMS Member Price: $30.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1411999; 131 pp
    MSC: Primary 20;

    Abstract. Let \(G\) be a simple algebraic group of exceptional type over an algebraically closed field of characteristic \(p\). Under some mild restrictions on \(p\), we classify all conjugacy classes of closed connected subgroups \(X\) of type \(A_1\); for each such class of subgroups, we also determine the connected centralizer and the composition factors in the action on the Lie algebra \({\mathcal L}(G)\) of \(G\). Moreover, we show that \({\mathcal L}(C_G(X))=C_{{\mathcal L}(G)}(X)\) for each subgroup \(X\). These results build upon recent work of Liebeck and Seitz, who have provided similar detailed information for closed connected subgroups of rank at least \(2\).

    In addition, for any such subgroup \(X\) we identify the unipotent class \({\mathcal C}\) meeting it. Liebeck and Seitz proved that the labelled diagram of \(X\), obtained by considering the weights in the action of a maximal torus of \(X\) on \({\mathcal L}(G)\), determines the (\(\mathrm{Aut}\,G\))-conjugacy class of \(X\). We show that in almost all cases the labelled diagram of the class \({\mathcal C}\) may easily be obtained from that of \(X\); furthermore, if \({\mathcal C}\) is a conjugacy class of elements of order \(p\), we establish the existence of a subgroup \(X\) meeting \({\mathcal C}\) and having the same labelled diagram as \({\mathcal C}\).

    Readership

    Graduate students and research mathematicians interested in group theory and generalizations.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Labelled diagrams
    • 3. Essential embeddings
    • 4. Unipotent classes
    • 5. Centralizers
    • 6. Results
    • 7. (Aut $G$)-conjugacy
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Volume: 1411999; 131 pp
MSC: Primary 20;

Abstract. Let \(G\) be a simple algebraic group of exceptional type over an algebraically closed field of characteristic \(p\). Under some mild restrictions on \(p\), we classify all conjugacy classes of closed connected subgroups \(X\) of type \(A_1\); for each such class of subgroups, we also determine the connected centralizer and the composition factors in the action on the Lie algebra \({\mathcal L}(G)\) of \(G\). Moreover, we show that \({\mathcal L}(C_G(X))=C_{{\mathcal L}(G)}(X)\) for each subgroup \(X\). These results build upon recent work of Liebeck and Seitz, who have provided similar detailed information for closed connected subgroups of rank at least \(2\).

In addition, for any such subgroup \(X\) we identify the unipotent class \({\mathcal C}\) meeting it. Liebeck and Seitz proved that the labelled diagram of \(X\), obtained by considering the weights in the action of a maximal torus of \(X\) on \({\mathcal L}(G)\), determines the (\(\mathrm{Aut}\,G\))-conjugacy class of \(X\). We show that in almost all cases the labelled diagram of the class \({\mathcal C}\) may easily be obtained from that of \(X\); furthermore, if \({\mathcal C}\) is a conjugacy class of elements of order \(p\), we establish the existence of a subgroup \(X\) meeting \({\mathcal C}\) and having the same labelled diagram as \({\mathcal C}\).

Readership

Graduate students and research mathematicians interested in group theory and generalizations.

  • Chapters
  • 1. Introduction
  • 2. Labelled diagrams
  • 3. Essential embeddings
  • 4. Unipotent classes
  • 5. Centralizers
  • 6. Results
  • 7. (Aut $G$)-conjugacy
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