On $p$-local factorization in a factorized finite group

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago Chicago, USA

Abstract

Factorizations play an important role in the analysis of the structure of finite groups. In this short note we extend a basic $p$-local Factorization Theorem in the textbook The Theory of Finite Groups by H. Kurzweil and B. Stellmacher that is applied in their Signalizer Functor Chapter. Our result is somewhat surprising and immediately implies their Factorization Theorem and should have other important applications. The proof of this basic result is tricky and elementary and could have been proved several years ago. The proof requires only well known facts about finite group factorizations and Sylow $p$-subgruops of finite groups. It resulted from the goal of simplifying the proof of the $p$-Local Factorization Theorem in the excellent H. Kurzweil and B. Stellmacher textbook Theory of Finite Groups. Hopefully this elementary result will inspire and lead to new important results on the structure of finite proofs, as is illustrated by the application to the Signalizer Functor Chapter in the textbook.

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References

[1] H. Kurzweil and B. Stellmacher, The theory of finite groups: an introduction, translated from the 1998 German original, Springer-Verlag, New York, 2004.
International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 11 May 2025

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History

  • Receive Date: 28 March 2024
  • Revise Date: 02 May 2025
  • Accept Date: 05 May 2025
  • Published Online: 11 May 2025

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