Partial groups, pregroups and realisability of fusion systems (Q6612189)

A pregroup is a set \(P\) equipped with a binary product \(m : D\rightarrow P\) , where \(D \subseteq P \times P\), subject to some group-like axioms (this concept was introduced by \textit{J. Stallings} [Group theory and three-dimensional manifolds. New Haven-London: Yale University Press. (1971; Zbl 0241.57001)]). For other purposes, \textit{A. Chermak} [Acta Math. 211, No. 1, 47-139 (2013; Zbl 1295.20021)] introduced another generalization of the groups, namely partial groups.\N\NIn this paper, the authors compare these two different notions of partially defined group structures. In particular they prove that the category of pregroups can be seen as a full subcategory of the category of partial groups. They also bring out some conjugation properties about elements and subgroups of finite order in pregroups and their universal groups. The authors then use these to investigate the question of realizability of fusion systems in finite pregroups.