Abstract. A p-local finite group consists of a finite p-group S, togetherwith a pair of categories which encode “conjugacy ” relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group. A p-local finite group consists of a finite p-group S, together with a pair of categories (F, L), of which F is modeled on the conjugacy (or fusion) in a Sylow subgroup of a finite group. The category L is essentially an extension of F and contains just enough extra information so that its p-completed nerve has many of the same properties as p-completed classifying spaces of finite groups. We recall the precise definitions of these objects in Section 1, and refer to [BLO2] and [5A1] for motivation for their study
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