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The Reductive Subgroups of $F_4$
 
David I. Stewart New College, Oxford, United Kingdom
Front Cover for The Reductive Subgroups of $F_4$
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
Front Cover for The Reductive Subgroups of $F_4$
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  • Back Cover for The Reductive Subgroups of $F_4$
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.

  • Table of Contents
     
     
    • 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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