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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
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: $30.60 Click above image for expanded view$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 ${\mathcal C}$.

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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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 ${\mathcal C}$.

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
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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