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

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 Q <= S, 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 -lo cal 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. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).