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Extension problems of morphisms in fusion systems. (Chinese. English summary) Zbl 1363.20016

Summary: An exploration was made on some equivalent descriptions and basic properties of the domestic subgroups and essential subgroups of a fusion system. A characterization was presented on maximal Sylow intersections and domestic subgroups. Precisely, the following result is proved: Let \(F\) be a saturated fusion system on a finite \(p\)-group \(P\). A fully normalized subgroup \(Q\) is \(F\)-essential if and only if \(Q\) has a strong non-extendible \(F\)-automorphism; \(Q\) is a domestic subgroup if and only if \(Q\) has a non-extendible \(F\)-automorphism.

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D15 Finite nilpotent groups, \(p\)-groups
20D45 Automorphisms of abstract finite groups
20D25 Special subgroups (Frattini, Fitting, etc.)
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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