# The Reductive Subgroups of \(F_{4}\)

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*David I. Stewart*

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

# Table of Contents

## The Reductive Subgroups of $F_{4}$

- Chapter 1. Introduction 18 free
- Chapter 2. Overview 714 free
- Chapter 3. General Theory 1118
- Chapter 4. Reductive subgroups of 𝐹₄ 2936
- 4.1. Preliminaries 2936
- 4.2. 𝐿-irreducible subgroups of 𝐿’ 3037
- 4.3. Cohomology of unipotent radicals 3138
- 4.4. Simple subgroups of parabolics: Proof of Theorem 1 3643
- 4.4.1. Subgroups of rank ≥2 3643
- 4.4.2. Subgroups of type 𝐴₁, 𝑝=3 4148
- 4.4.2.1. (r,s,t)=(0,0,1) 4451
- 4.4.2.2. (r,s,t)=(1,1,0) 4653
- 4.4.2.3. (r,s,t)=(0,2,1) 4653
- 4.4.2.4. (r,s,t)=(0,s,s-1), 𝑠>2, (1,s,0) 𝑠>3, (1,3,0) 4754
- 4.4.2.5. (r,s,t)=(0,1,1), (0,s,1) 𝑠>2, (0,2,3), (0,s,s+1) 𝑠>2 4754
- 4.4.2.6. (r,s,t)=(0,1,0) 4754
- 4.4.2.7. (r,s,t)=(r,r+1,0) (r>1), (0,1,t) (t>2) 4855
- 4.4.2.8. (r,s,t)=(0,1,2) 4855
- 4.4.2.9. (r,s,t)=(1,2,0) 4855

- 4.4.3. Subgroups of type 𝐴₁, 𝑝=2 4956

- 4.5. Non-simple yet semisimple subgroups of parabolics: Proof of Theorem 2 6269
- 4.6. Subgroups not in parabolics: Proof of Theorem 4 6875
- 4.7. Proofs of Corollaries 3, 5, 6, 7 and 8 6976

- Chapter 5. Appendices 7582
- Bibliography 8794