Finite localities I

Abstract

This is the first of a series of papers concerning what might be thought of as ‘locally grouped spaces’, in a loose analogy with the locally ringed spaces of algebraic geometry. The spaces that we have in mind are simplicial sets that generalise the simplicial sets that underlie and determine the classifying spaces of finite (or compact) groups. If the analogy is pursued, then the role of ‘structure sheaf’ is provided by the ‘fusion systems’ associated with these spaces. Our approach here will be purely algebraic and combinatorial, so we will not be concerned with topological realisations. All of the groups to be considered will be finite; but a parallel series of papers representing some joint work with Alex Gonzalez will considerably broaden the scope.

Kernels of localities

“…Properties (a),(b) and (c) correspond to the statements (a),(b) and (c) in [8, Lemma 2.3] except for the fact stated in (b) that , which is however clearly true if one uses the definition of a locality above. Property (d) holds by [8, Proposition 2.5(a), (b)], and property (e) is stated in [8, Corollary 2.6]. Property (f) follows from (e) and (c).…”

“…In this section, we introduce some basic definitions and notations that will be used in the remainder of the paper. We refer the reader to [8] for a more comprehensive introduction to partial groups and localities.…”

“…By the Frattini Lemma [8, Corollary 3.11], there exist and such that , and f is -maximal (in the sense of [8, Definition 3.6]). By [8, Proposition 3.9] and then [8, Lemma 3.1(a)], every -maximal element is in . Moreover, the Splitting Lemma [8, Lemma 3.12] gives .…”

“…By [8, Proposition 3.9] and then [8, Lemma 3.1(a)], every -maximal element is in . Moreover, the Splitting Lemma [8, Lemma 3.12] gives .…”

“…Observe that T is an element of , so is a subgroup of . It follows therefore from [8, Theorem 4.3(b), Corollary 4.5] that is a group. Thus, can be seen as an extension of the group by the locality .…”