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Finite Groups Which are Almost Groups of Lie Type in Characteristic $\mathbf p$

About this Title

Chris Parker, Gerald Pientka, Andreas Seidel and Gernot Stroth

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 292, Number 1452
ISBNs: 978-1-4704-6729-6 (print); 978-1-4704-7697-7 (online)
DOI: https://doi.org/10.1090/memo/1452
Published electronically: November 17, 2023
Keywords: Finite groups, embedding, identification of finite simple groups

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Table of Contents

Chapters

• 1. Introduction
• 2. Preliminary Group Theoretical Results
• 3. Identification Theorems of Some Almost Simple Groups
• 4. Strongly $\mathbf {p}$-embedded Subgroups
• 5. Sylow Embedded Subgroups of Linear Groups
• 6. Main Hypothesis and Notation for the Proof of the Main Theorems
• 7. The Embedding of $\mathbf Q$ in $\mathbf G$ Under Hypothesis
• 8. The Groups Which Satisfy Hypothesis with $\mathbf { N_{F^*(H)}(Q)}$ Not Soluble and $\mathbf {N_G(Q) \not \le H}$
• 9. The Groups with $\mathbf {F^*(H)\cong PSL_3(p^e)}$, $\mathbf p$ Odd
• 10. The Groups with $\mathbf {F^*(H) \cong PSL_3(2^e)}$ or $\mathbf {Sp_4(2^e)’}$
• 11. The Groups with $\mathbf {F^*(H) \cong \mathbf {Sp}_{2n}(2^e)}$, $\mathbf {n \ge 3}$
• 12. The Groups with $\mathbf {F^*(H)\cong {}^2F_4(2^{2e+1})^\prime }$
• 13. The Groups with $\mathbf {F^*(H)\cong F_4(2^e)}$
• 14. The Case When $\mathbf {p = 2}$ and Centralizer of Some $\mathbf {2}$-central Element of $\mathbf {H}$ is Soluble
• 15. The Groups with $\mathbf {F^*(H) \cong G_2(3^e)}$
• 16. The Groups with $\mathbf {F^*(H)\cong P\Omega ^+_8(3)}$ and $\mathbf {N_G(Q) \not \le H}$
• 17. The Case When $\mathbf {p = 3}$, the Centralizer of Some $\mathbf {3}$-central Element of $\mathbf {H}$ is Soluble and $\mathbf {N_G(Q) \not \le H}$
• 18. Proof of Theorem and Theorem
• 19. Groups Which Satisfy Hypothesis with $\mathbf {N_G(Q) \le H}$ and Some $\mathbf {p}$-local Subgroup Containing $\mathbf {S}$ Not Contained in $\mathbf {H}$
• 20. Proof of Theorem
• 21. Proof of Theorem
• 22. Proof of Main Theorem  and Main Theorem
• A. Properties of Finite Simple Groups of Lie Type
• B. Properties Alternating Groups
• C. Small Modules for Finite Simple Groups
• D. $\mathbf {p}$-local Properties of Groups of Lie Type in Characteristic $\mathbf {p}$
• E. Miscellanea

Abstract

Let $p$ be a prime. In this paper we investigate finite $\mathcal K_{\{2,p\}}$-groups $G$ which have a subgroup $H \le G$ such that $K \le H = N_G(K) \le \operatorname {Aut}(K)$ for $K$ a simple group of Lie type in characteristic $p$, and $|G:H|$ is coprime to $p$. If $G$ is of local characteristic $p$, then $G$ is called almost of Lie type in characteristic $p$. Here $G$ is of local characteristic $p$ means that for all nontrivial $p$-subgroups $P$ of $G$, and $Q$ the largest normal $p$-subgroup in $N_G(P)$ we have the containment $C_G(Q)\le Q$. We determine details of the structure of groups which are almost of Lie type in characteristic $p$. In particular, in the case that the rank of $K$ is at least $3$ we prove that $G = H$. If $H$ has rank $2$ and $K$ is not $\operatorname {PSL}_3(p)$ we determine all the examples where $G \ne H$. We further investigate the situation above in which $G$ is of parabolic characteristic $p$. This is a weaker assumption than local characteristic $p$. In this case, especially when $p \in \{2,3\}$, many more examples appear.

In the appendices we compile a catalogue of results about the simple groups with proofs. These results may be of independent interest.