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NONREALIZABILITY OF CERTAIN REPRESENTATIONS IN FUSION SYSTEMS

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

Published online by Cambridge University Press:  11 April 2023

BOB OLIVER
BOB OLIVER*
Affiliation:
Université Sorbonne Paris Nord, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément, Villetaneuse 93430, France
*
e-mail: [email protected]
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Abstract

For a finite abelian p-group A and a subgroup $\Gamma \le \operatorname {\mathrm {Aut}}(A)$, we say that the pair $(\Gamma ,A)$ is fusion realizable if there is a saturated fusion system ${\mathcal {F}}$ over a finite p-group $S\ge A$ such that $C_S(A)=A$, $\operatorname {\mathrm {Aut}}_{{\mathcal {F}}}(A)=\Gamma $ as subgroups of $\operatorname {\mathrm {Aut}}(A)$, and . In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\Gamma $ one of the Mathieu groups, that the only ${\mathbb {F}}_p\Gamma $-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.

Keywords

finite groupsSylow subgroupsfusionfinite simple groupsmodular representations

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
Journal of the Australian Mathematical Society , Volume 116 , Issue 2 , April 2024 , pp. 257 - 288
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Michael Giudici

B. Oliver is partially supported by UMR 7539 of the CNRS. Part of this work was carried out at the Isaac Newton Institute for Mathematical Sciences during the GRA2 programme, supported by EPSRC Grant No. EP/K032208/1.

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