A generalization of Alperin Fusion theorem and its applications

Let F be a saturated fusion system on a finite p-group S, and let Pbe a strongly F-closed subgroup of S. We define the concept "F-essential subgroups with respect to P" which are some proper subgroups of Psatisfying some technical conditions and show that an F-isomorphism between subgroups of Pcan be factorizedby some automorphisms of Pand F-essential subgroups with respect to P. When Pis taken to be equal to S, the Alperin-Goldschmidt fusion theorem can be obtained as a special case. We also show that P (sic) F if and only if there is no F-essential subgroup with respect to P. The following definition is made: A p-group Pis strongly resistantin saturated fusion systems if P (sic) F whenever there is an over p-group Sand a saturated fusion system F on Ssuch that Pis strongly F-closed. It is shown that several classes of p-groups are strongly resistant, which appears as our third main theorem. We also give a new necessary and sufficient criterion for a strongly F-closed subgroup to be normal in F. These results are obtained as a consequence of developing a theory of quasi and semi-saturated fusion systems, which seems to be interesting in its own right. (c) 2024 The Author(s). Published by Elsevier Inc.