Commuting partial normal subgroups and regular localities

Abstract

In this paper, important concepts from finite group theory are translated to localities, in particular to linking localities. Here localities are group-like structures associated to fusion systems which were introduced by Chermak. Linking localities (by Chermak also called proper localities) are special kinds of localities which correspond to linking systems. Thus they contain the algebraic information that is needed to study $p$-completed classifying spaces of fusion systems as generalizations of $p$-completed classifying spaces of finite groups. Because of the group-like nature of localities, there is a natural notion of partial normal subgroups. Given a locality $\mathcal{L}$ and a partial normal subgroup $\mathcal{N}$ of $\mathcal{L}$, we show that there is a largest partial normal subgroup $\mathcal{N}^\perp$ of $\mathcal{L}$ which, in a certain sense, commutes elementwise with $\mathcal{N}$ and thus morally plays the role of a "centralizer" of $\mathcal{N}$ in $\mathcal{L}$. This leads to a nice notion of the generalized Fitting subgroup $F^*(\mathcal{L})$ of a linking locality $\mathcal{L}$. Building on these results we define and study special kinds of linking localities called regular localities. It turns out that there is a theory of components of regular localities akin to the theory of components of finite groups. The main concepts we introduce and work with in the present paper (in particular $\mathcal{N}^\perp$ in the special case of linking localities, $F^*(\mathcal{L})$, regular localities and components of regular localities) were already introduced and studied in a preprint by Chermak. However, we give a different and self-contained approach to the subject where we reprove Chermak's theorems and also show several new results.