A generalization of Alperin Fusion theorem and its applications

Let F be a saturated fusion system on a finite p-group S, and let P be a strongly F-closed subgroup of S. We define the concept "F-essential subgroups with respect to P" which are some proper subgroups of P satisfying some technical conditions and show that an F-isomorphism between subgroups of P can be factorised by some automorphisms of P and F-essential subgroups with respect to P. When P is taken to be equal to S, the Alperin-Goldschmidt fusion theorem can be obtained as a special case. We also show that P⊴F if and only if there is no F-essential subgroup with respect to P. The following definition is made: A p-group P is strongly resistant in saturated fusion systems if P⊴F whenever there is an over p-group S and a saturated fusion system F on S such that P is 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.