A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver

摘要

We extend the notion of a p-local finite group (defined in BLO03) to the notion of a p-local group. We define morphisms of p-local groups, obtaining thereby a category, and we introduce the notion of a representation of a p-local group via signalizer functors associated with groups. We construct a chain G = (G_0 →G_1 →...) of 2-local finite groups, via a representation of a chain G* = (G_0 → G_1 →...) of groups, such that G0 is the 2-local finite group of the third Conway sporadic group Co_3, and for n > 0, G_n is one of the 2-local finite groups constructed by Levi and Oliver in LO02. We show that the direct limit G of G exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain G*. The 2-completed classifying space of G is shown to be the classifying space B DI(4) of the exotic 2-compact group of Dwyer and Wilkerson DW93.