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On $p$-local factorization in a factorized finite group | ||
| International Journal of Group Theory | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 21 اردیبهشت 1404 اصل مقاله (322.3 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2025.141044.1899 | ||
| نویسنده | ||
| Morton E. Harris* | ||
| Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago Chicago, USA | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Factorized Finite Groups؛ $p$-Local Factorization؛ covering $p$-group normalizer | ||
| مراجع | ||
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[1] H. Kurzweil and B. Stellmacher, The theory of finite groups: an introduction, translated from the 1998 German original, Springer-Verlag, New York, 2004. | ||
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