Abstract
We show a classification method for finite groupoids using a decomposition of its connected components on a product of a coarse groupoid and of an isotropy group. Moreover, we discuss the cardinality of cosets and its relation with the index. We extend Lagrange’s Theorem by relating the order of a groupoid with the order of a subgroupoid and its index, highlighting the connected and wide cases. We define characteristic subgroupoids and we show some of its basic properties. Finally, we establish a Sylow theory for groupoids aiming to prove a converse for Lagrange’s Theorem, that is, given a finite groupoid and a positive integer n, is there a subgroupoid with order n? The answer depends on the integer n and it can be used to generalize all three Sylow Theorems for the case of groupoids.
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Communicated by Francisco Cesar Polcino Milies.
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Beier, G., Garcia, C., Lautenschlaeger, W.G. et al. Generalizations of Lagrange and Sylow theorems for groupoids. São Paulo J. Math. Sci. 17, 720–739 (2023). https://doi.org/10.1007/s40863-022-00351-7
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DOI: https://doi.org/10.1007/s40863-022-00351-7