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Reduced fusion systems over p-groups with abelian subgroup of index p: III

Part of: Structure and classification of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  30 January 2019

Bob Oliver  and
Albert Ruiz
Bob Oliver
Affiliation:
LAGA, Institut Galilée, Av. J-B Clément, 93430Villetaneuse, France ([email protected])
Albert Ruiz
Affiliation:
Departament de Mathemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain ([email protected])
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Abstract

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all 𝔽pG-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [15] using the classification of finite simple groups.

Keywords

finite groupsfusionfinite simple groupsmodular representationsSylow subgroups

MSC classification

Primary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Secondary: 20C20: Modular representations and characters 20D05: Finite simple groups and their classification 20E45: Conjugacy classes
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1187 - 1239
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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