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In this article, we compare two different notions of partially defined group structures, namely partial groups and pregroups, as introduced by Chermak [4] and Stallings [18] respectively. In particular we prove that the category of pregroups can be seen as a full subcategory of the category of partial groups. We also bring out some conjugation properties about elements and subgroups of finite order in pregroups and their universal groups. We then use these to investigate the question of realisability of fusion systems in finite pregroups. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of linking systems associated to saturated fusion systems over discrete p -toral groups were proved by Levi and Libman. Their proof makes indirect use of the classification of the finite simple groups. Here we extend some results and arguments of Glauberman and Lynd to show how to remove this assumption. [ABSTRACT FROM AUTHOR]

We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system F . More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of F -stable elements to F c -stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group. We then study the problem of inducing an idempotent from an F -characteristic ( S , S ) -biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by F c -stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a p -local finite group can be computed by these F c -stable elements. [ABSTRACT FROM AUTHOR]

These notes are defining the notion of centric linking system for a locally finite group. If a locally finite group G has countable Sylow p -subgroups, we prove that, with a finiteness condition on the set of intersections of Sylow p -subgroups, the p -completion of its classifying space is homotopy equivalent to the p -completion of the nerve of its centric linking system. [ABSTRACT FROM AUTHOR]

Nucleation phenomena commonly observed in our every day life are of fundamental, technological and societal importance in many areas, but some of their most intimate mechanisms remain however to be unravelled. Crystal nucleation, the early stages where the liquid-to-solid transition occurs upon undercooling, initiates at the atomic level on nanometre length and sub-picoseconds time scales and involves complex multidimensional mechanisms with local symmetry breaking that can hardly be observed experimentally in the very details. To reveal their structural features in simulations without a priori, an unsupervised learning approach founded on topological descriptors loaned from persistent homology concepts is proposed. Applied here to monatomic metals, it shows that both translational and orientational ordering always come into play simultaneously as a result of the strong bonding when homogeneous nucleation starts in regions with low five-fold symmetry. It also reveals the specificity of the nucleation pathways depending on the element considered, with features beyond the hypothesis of Classical Nucleation Theory. [ABSTRACT FROM AUTHOR]