Reduced fusion systems over 2-groups of small order

We prove, when \(S\) is a \(2\)-group of order at most \(2^9\), that each reduced fusion system over \(S\) is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a \(2\)-group of order at most \(2^9\) is realizable. What is most interesting...

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Published in:arXiv.org 2017-05
Main Authors: Andersen, Kasper K S, Oliver, Bob, Ventura, Joana
Format: Article
Language:English
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Summary:We prove, when \(S\) is a \(2\)-group of order at most \(2^9\), that each reduced fusion system over \(S\) is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a \(2\)-group of order at most \(2^9\) is realizable. What is most interesting about this result is the method of proof: we show that among \(2\)-groups with order in this range, the ones which can be Sylow \(2\)-subgroups of finite simple groups are almost completely determined by criteria based on Bender's classification of groups with strongly \(2\)-embedded subgroups.
ISSN:2331-8422