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The Local Structure Theorem for Finite Groups with a Large $p$-Subgroup


About this Title

U. Meierfrankenfeld, B. Stellmacher and G. Stroth

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 242, Number 1147
ISBNs: 978-1-4704-1877-9 (print); 978-1-4704-2948-5 (online)
DOI: http://dx.doi.org/10.1090/memo/1147
Published electronically: April 8, 2016
Previous Version: Original version posted April 8, 2016 with incorrect title
Keywords:Local characteristic $p$, large subgroup, finite simple groups

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Table of Contents


Chapters

  • Introduction
  • Chapter 1. Definitions and Preliminary Results
  • Chapter 2. The Case Subdivision and Preliminary Results
  • Chapter 3. The Orthogonal Groups
  • Chapter 4. The Symmetric Case
  • Chapter 5. The Short Asymmetric Case
  • Chapter 6. The Tall $char\, p$-Short Asymmetric Case
  • Chapter 7. The $char\, p$-Tall $Q$-Short Asymmetric Case
  • Chapter 8. The $Q$-Tall Asymmetric Case I
  • Chapter 9. The $Q$-tall Asymmetric Case II
  • Chapter 10. Proof of the Local Structure Theorem
  • Appendix A. Module theoretic Definitions and Results
  • Appendix B. Classical Spaces and Classical Groups
  • Appendix C. FF-Module Theorems and Related Results
  • Appendix D. The Fitting Submodule
  • Appendix E. The Amalgam Method
  • Bibliography

Abstract


Let $p$ be a prime, $G$ a finite $\mathcal K_{p}$-group $S$ a Sylow $p$-subgroup of $G$ and $Q$ a large subgroup of $G$ in $S$ (i.e., $C_{G}(Q) ≤Q$ and $N_{G}(U) ≤N_{G}(Q)$ for $1 ≠U ≤C_{G}(Q)$). Let $L$ be any subgroup of $G$ with $S≤L$, $O_{p}(L)≠1$ and $Q⋬L$. In this paper we determine the action of $L$ on the largest elementary abelian normal $p$-reduced $p$-subgroup $Y_{L}$ of $L$.

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