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Subgroup families controlling $$p$$-local finite groups. (English) Zbl 1090.20026
Let $$p$$ be a prime, $$G$$ a finite group and $$S$$ a Sylow $$p$$-subgroup of $$G$$. A $$p$$-fusion category for $$G$$ is a category $${\mathcal F}={\mathcal F}^{\mathcal H}_S(G)$$, whose object set is a set $$\mathcal H$$ of subgroups of $$S$$, and whose morphisms are the homomorphisms between subgroups in $$\mathcal H$$ induced by conjugation in $$G$$. The associated linking category $${\mathcal L}={\mathcal L}^{\mathcal H}_S(G)$$ has the same objects, and morphisms from $$P$$ to $$Q$$ are given by the formula $$\text{Mor}_{\mathcal L}(P,Q)=\{x\in G\mid xPx^{-1}\leq Q\}/O^p(C_G(P))$$, where, $$O^p(-)$$ is the subgroup generated by elements of order prime to $$p$$.
It was shown by C. Broto, R. Levi and B. Oliver [Invent. Math. 151, No. 3, 611-664 (2003; Zbl 1042.55008)] that the homotopy theory of the nerve $$|{\mathcal L}^{\mathcal H}_S(G)|$$ (for the right choice of $$\mathcal H$$) is closely related to the $$p$$-local homotopy theory of $$BG$$.
The main goal of this paper is to examine the role of the set $$\mathcal H$$ of subgroups of $$S$$ on which the fusion and linking systems are defined; that is, to show when the set can be changed without changing $$\mathcal F$$ and $$\mathcal L$$ in an ‘essential’ way.
The main results are the following: Theorem A. Let $$\mathcal F$$ be a fusion system over a finite $$p$$-group $$S$$. Let $$\mathcal H$$ be a set of subgroups of $$S$$ closed under $$\mathcal F$$-conjugacy, with the property that each $$\mathcal F$$-centric subgroup of $$S$$ not in $$\mathcal H$$ is $$\mathcal F$$-conjugate to some subgroup $$P\leq S$$ such that $$\text{Out}_S(P)\cap O_p({\text{Out}}_{\mathcal F}(P))\neq 1$$. Assume that all morphisms in $$\mathcal F$$ are composites of restrictions of morphisms between subgroups in $$\mathcal H$$. If $$\mathcal F$$ satisfies the axioms of saturation when applied to subgroups of $$S$$ in $$\mathcal H$$, then $$\mathcal F$$ is saturated.
When $$\mathcal H$$ is the set of all $$\mathcal F$$-centric subgroups of $$S$$, then it is due to L. Puig [Unpublished notes, CNRS, Université de Paris 7, ca. 1990, Theorem 1.17]. Theorem A can be thought of as a converse to Alperin’s fusion theorem for abstract fusion systems (as shown by L. Puig [loc. cit.] and C. Broto, R. Levi and B. Oliver [J. Am. Math. Soc. 16, No. 4, 779-856 (2003; Zbl 1033.55010), Theorem A.10]), which says that if $$\mathcal F$$ is a saturated fusion system, then it is generated by restrictions of automorphisms of $$\mathcal F$$-centric $$\mathcal F$$-radical subgroups.
Theorem B. Let $$(S,{\mathcal F},{\mathcal L})$$ be a $$p$$-local finite group, i.e. $$S$$ is a finite $$p$$-group, $$\mathcal F$$ is a saturated fusion system over $$S$$, and $$\mathcal L$$ is a centric linking system associated to $$\mathcal F$$. Then there exists a category $${\mathcal L}^q$$ containing $$\mathcal L$$ as a full subcategory, whose objects are the $$\mathcal F$$-quasicentric subgroups of $$S$$, and such that the inclusion of nerves $$|{\mathcal L}|\subseteq|{\mathcal L}^q|$$ is a homotopy equivalence. Furthermore, if $$\mathcal H$$ is any collection of $$\mathcal F$$-quasicentric subgroups of $$S$$ containing all $$P\leq S$$ which are both $$\mathcal F$$-centric and $$\mathcal F$$-radical, and $${\mathcal L}^{\mathcal H}\subseteq{\mathcal L}^q$$ is the full subcategory whose objects are the subgroups in $$\mathcal H$$, then the inclusions of $${\mathcal L}^{\mathcal H}$$ and $$\mathcal L$$ in $${\mathcal L}^q$$ induce homotopy equivalences $$|{\mathcal L}^{\mathcal H}|\simeq|{\mathcal L}^q|\simeq|{\mathcal L}|$$.
The paper concludes with a specialized family of examples: fusion systems whose entire structure is controlled by a single $$p$$-subgroup.

MSC:
 20J15 Category of groups 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20C20 Modular representations and characters 20D30 Series and lattices of subgroups
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