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The Reductive Subgroups of $F_4$

David I. Stewart New College, Oxford, United Kingdom
Available Formats:
Electronic ISBN: 978-0-8218-9873-4
Product Code: MEMO/223/1049.E
88 pp
List Price: $69.00 MAA Member Price:$62.10
AMS Member Price: $41.40 Click above image for expanded view The Reductive Subgroups of$F_4$David I. Stewart New College, Oxford, United Kingdom Available Formats:  Electronic ISBN: 978-0-8218-9873-4 Product Code: MEMO/223/1049.E 88 pp  List Price:$69.00 MAA Member Price: $62.10 AMS Member Price:$41.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 2232013
MSC: Primary 20; 18;

Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p\geq 0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no proper parabolic subgroup of $G$; and $G$-reducible if it is in some proper parabolic of $G$. In this paper, the author considers the case that $G=F_4(K)$.

The author finds all conjugacy classes of closed, connected, semisimple $G$-reducible subgroups $X$ of $G$. Thus he also finds all non-$G$-completely reducible closed, connected, semisimple subgroups of $G$. When $X$ is closed, connected and simple of rank at least two, he finds all conjugacy classes of $G$-irreducible subgroups $X$ of $G$. Together with the work of Amende classifying irreducible subgroups of type $A_1$ this gives a complete classification of the simple subgroups of $G$.

The author also uses this classification to find all subgroups of $G=F_4$ which are generated by short root elements of $G$, by utilising and extending the results of Liebeck and Seitz.

• Chapters
• 1. Introduction
• 2. Overview
• 3. General Theory
• 4. Reductive subgroups of $F_4$
• 5. Appendices
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Volume: 2232013
MSC: Primary 20; 18;

Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p\geq 0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no proper parabolic subgroup of $G$; and $G$-reducible if it is in some proper parabolic of $G$. In this paper, the author considers the case that $G=F_4(K)$.

The author finds all conjugacy classes of closed, connected, semisimple $G$-reducible subgroups $X$ of $G$. Thus he also finds all non-$G$-completely reducible closed, connected, semisimple subgroups of $G$. When $X$ is closed, connected and simple of rank at least two, he finds all conjugacy classes of $G$-irreducible subgroups $X$ of $G$. Together with the work of Amende classifying irreducible subgroups of type $A_1$ this gives a complete classification of the simple subgroups of $G$.

The author also uses this classification to find all subgroups of $G=F_4$ which are generated by short root elements of $G$, by utilising and extending the results of Liebeck and Seitz.

• Chapters
• 1. Introduction
• 2. Overview
• 3. General Theory
• 4. Reductive subgroups of $F_4$
• 5. Appendices
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