Fusion systems and localities. (English) Zbl 1295.20021

Chermak, Andrew

Acta Math. 211, No. 1, 47-139 (2013).

Fusion systems generalize fusion in a Sylow \(p\)-subgroup \(P\) of a group \(G\) given by conjugacy, i.e., inner automorphisms of \(G\). Most important are so-called saturated fusion systems. All those, which are constructed as above are saturated. On the other hand not all saturated fusions system are the fusion system of a properly chosen group. These usually are called exotic and many examples for odd \(p\) are known. Nevertheless for so-called constrained fusion systems there is a uniquely determined model.
In this paper now the author proves that there is also a uniquely determined model for arbitrary saturated fusion systems, which is called a linking system. This solves a long standing open question. Precisely: Let \(\mathcal F\) be a saturated fusion system on the finite \(p\)-group \(P\), \(p\) a prime. Then there exists a centric linking system \(\mathcal L\) such that \(\mathcal F\) is the fusion system generated by the conjugation maps in \(\mathcal L\) between subgroups of \(P\). Further \(\mathcal L\) is uniquely determined by \(\mathcal F\), up to isomorphisms which restrict to the identity on \(P\).
For the proof the author introduces partial groups and localities which might be of independent interest. A partial group basically is like a group in which multiplication is not always defined according to some rules. A partial group \(\mathcal M\) together with a set \(\Delta\) of subgroups which control multiplication is called an objective partial group. If our \(p\)-group \(P\) in question is the unique maximal element in \(\Delta\), then \((\mathcal M,\Delta,P)\) is called a locality. A locality is a \(\Delta\)-linking system if for all \(U\in\Delta\), \(C_{\mathcal L}(U)=Z(U)\). If \(\Delta\) is the set of all \(U\leq P\) such that \(C_P(Q)=Z(Q)\) for every \(\mathcal L\)-conjugate \(Q\) of \(U\), \(Q\leq P\), then \(\mathcal L\) is called a centric linking system. With any locality \(\mathcal L=(\mathcal M,\Delta,P)\) is associated a fusion system \(\mathcal F=\mathcal F_P(\mathcal L)\) on \(P\). The theorem basically says that all saturated fusion systems arise in this way.
The proof uses the Thompson-subgroup \(J(R)\) of subgroups \(R\) of \(P\) and so it uses failure of factorization and the classification of such modules. This is where the paper is dependent on the classification of the finite simple groups.
There is a different proof of the same theorem due to B. Oliver [in Acta Math. 211, No. 1, 141-175 (2013; Zbl 1292.55007)].

Reviewer: Gernot Stroth (Halle)

MSC:

20D30	Series and lattices of subgroups
20D20	Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
55R35	Classifying spaces of groups and \(H\)-spaces in algebraic topology

Keywords:

objective partial groups; linking systems; \(p\)-local finite groups; localities; saturated fusion systems; finite \(p\)-groups

Citations:

Zbl 1292.55007

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