Autoparatopisms of quasigroups and Latin squares

Paratopism is a well known action of the wreath product Sn wr S3 on Latin squares of order n. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let Par(n) denote the set of paratopisms that are an autoparatopism of at least one Latin square of order n. We prove a number of general properties of autoparatopisms which between them are sufficient to determine Par(n) for n≤17.

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