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A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver. (English) Zbl 1213.20017
C. Broto, R. Levi and B. Oliver [J. Am. Math. Soc. 16, No. 4, 779-856 (2003; Zbl 1033.55010)] introduced the notion of a \(p\)-local finite group \(\mathcal S\), consisting of a finite \(p\)-group \(S\) and a pair of categories \(\mathcal F\) and \(\mathcal L\) (the fusion system and centric linking system) whose objects are subgroups of \(S\), and which satisfy axioms which reflect the structure of a finite group having \(S\) as Sylow \(p\)-subgroup. If \(G\) is a finite group with Sylow \(p\)-subgroup \(S\), then there is a canonical construction which associates to \(G\) a \(p\)-local finite group \(\mathcal S=\mathcal S_S(G)\), such that the \(p\)-completed nerve of \(\mathcal L\) is homotopically equivalent to the \(p\)-completed classifying space of \(G\). A \(p\)-local finite group \(\mathcal S\) is said to be exotic if \(\mathcal S\) is not equal to \(\mathcal S_S(G)\) for any finite group \(G\) with Sylow subgroup \(S\).
In the present paper the notion of a \(p\)-local finite group is extended to the notion of a \(p\)-local group. The authors define morphisms of \(p\)-local groups, obtaining thereby a category, and introduce the notion of a representation of a \(p\)-local group via signalizer functors associated with groups. They construct a chain \(\mathcal G=(\mathcal S_0\to\mathcal S_1\to\cdots)\) of 2-local finite groups, via a representation of a chain \(\mathcal G^*=(G_0\to G_1\to\cdots)\) of groups, such that \(\mathcal S_0\) is the 2-local finite group of the third Conway sporadic group \(Co_3\), and for \(n>0\), \(\mathcal S_n\) is one of the 2-local finite groups constructed by R. Levi and B. Oliver [in Geom. Topol. 6, 917-990 (2002; Zbl 1021.55010)]. The authors show that the direct limit \(\mathcal S\) of \(\mathcal G\) exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain \(\mathcal G^*\). The 2-completed classifying space of \(\mathcal S\) is shown to be the classifying space \(BDI(4)\) of the exotic 2-compact group of W. G. Dwyer and C. W. Wilkerson [J. Am. Math. Soc. 6, No. 1, 37-64 (1993; Zbl 0769.55007)].

MSC:
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D15 Finite nilpotent groups, \(p\)-groups
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55R37 Maps between classifying spaces in algebraic topology
20J15 Category of groups
55P99 Homotopy theory
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