All Sylow p-subgroups of G are conjugate. Moreover, every p-subgroup of G lies inside at least one Sylow p-subgroup. In the present context this can be used as follows.
This result allows to fuse certain -classes of p-subgroups into singletons.
A procedure which systematically searches for places where Corollary 4.8 can be applied in order to produce a stronger pre-fusion map will be called CheckSylowFrame.