The global structure theorem for finite groups with an abelian large p-subgroup

Ulrich Meierfrankenfeld, Chris Parker, Gernot Stroth*

*Corresponding author for this work

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Abstract

For a prime p, the Local Structure Theorem [15] studies finite groups G with the property that a Sylow p-subgroup S of G is contained in at least two maximal p-local subgroups. Under the additional assumptions that G contains a so called large p-subgroup QS , and that composition factors of the normalizers of non-trivial p-subgroups are from the list of the known simple groups, [15] partially describes the p-local subgroups of G containing S, which are not contained in NG(Q). In the Global Structure Theorem, we extend the work of [15] and describe NG(Q) and, in almost all cases, the isomorphism type of the almost simple subgroup H generated by the p-local over-groups of S in G. Furthermore, for p = 2 , the isomorphism type of G is determined. In this paper, we provide a reduction framework for the proof of the Global Structure Theorem and also prove the Global Structure Theorem when Q is abelian.
Original languageEnglish
Pages (from-to)174-215
Number of pages42
JournalJournal of Algebra
Volume640
Early online date14 Nov 2023
DOIs
Publication statusPublished - 15 Feb 2024

Bibliographical note

Acknowledgment:
We thank Bernd Stellmacher for numerous comments which have improved the paper. The third author was partially supported by the DFG STR 203/20-1.

Keywords

  • Finite simple groups

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