Intrinsic components in involution centralizers of fusion systems

This paper lays the foundation for the study of the saturated 2-fusion systems in which the centralizer of some fully centralized involution has a component whose center is nontrivial.The results in this paper are part of a program to, first, classify a large subclass of the class of simple 2-fusion systems of component type, and then, second, to use the theorem on fusion systems to simplify the proof of the theorem classifying the finite simple groups.See [4; 5] for a description of the program; there is also a bit of discussion of the program below.Let p be a prime and S a finite p-group.A fusion system on S is a category F whose objects are the subgroups of S and, for subgroups P, Q of S, the set hom F (P, Q) of morphisms from P to Q is a set of injective group homomorphisms from P to Q, and that set satisfies two weak axioms.The standard example is the fusion system F S (G) for G a finite group and S Syl p (G), whose morphisms are those induced via conjugation in G.A fusion system is saturated if it satisfies two more axioms easily seen to hold in the standard example using Sylow's theorem.See [12] for notation, terminology, and basic definitions and results on fusion systems.Let F be a saturated fusion system on a finite 2-group S. Proceeding by analogy with finite groups, one can define the notion of a normal subsystem of F, which can then be used to define the notions of simple and quasisimple systems, subnormal subsystems of F, and the set Comp(F) of components of F. For t an involution in S the centralizer C F (t) of t in F is defined, and if t is fully centralized (i.e.,Define C(F) to be the set of components of centralizers of involutions in F; that is, C C(F) if there exists some involution t S and a conjugate ( t, C) of (t, C) such that t is fully centralized and C Comp(C F ( t)); we write I(C) for the set of such involutions t.We say that F is of component type if C(F) is nonempty.