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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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