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

Abstract

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 $\mathfrak{G}=(\mathcal{G}_0\to\mathcal{G}_1\to\cdots)$ of $2$-local finite groups, via a representation of a chain $\frak G^*=(G_0\to G_1\to\cdots)$ of groups, such that $\mathcal{G}_0$ is the $2$-local finite group of the third Conway sporadic group ${\rm Co}_3$, and for $n>0$, $\mathcal{G}_n$ is one of the $2$-local finite groups constructed by Levi and Oliver in [LO02]. We show that the direct limit $\mathcal{G}$ of $\mathfrak{G}$ exists in the category of $2$-local groups, and that it is the $2$-local group of the union of the chain $\mathfrak{G}^*$. The $2$-completed classifying space of $\mathcal{G}$ is shown to be the classifying space $B\operatorname{D}\operatorname{I}(4)$ of the exotic $2$-compact group of Dwyer and Wilkerson [DW93].