Subgroup families controlling \(p\)-local finite groups.

*(English)*Zbl 1090.20026Let \(p\) 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When \(\mathcal H\) is the set of all \(\mathcal F\)-centric subgroups of \(S\), then it is due to L. Puig [Unpublished notes, CNRS, Université de Paris 7, ca. 1990, Theorem 1.17].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.

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.

Reviewer: Olympia Talelli (Athens)

##### 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 |