Autoparatopisms of Latin squares

Abstract In this thesis we study autoparatopisms and near-autoparatopisms of Latin squares. Also we find a family of Latin squares with an unique intercalate and no larger subsquares. Paratopism is a well known action of the wreath product Sn oS3 on Latin squares oforder 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. Suppose that n ≡ ±1 mod 6 and n ≥ 7. We construct a Latin square Ln of order n with the following properties: • Ln has no proper subsquares of order 3 or more. • Ln has exactly one intercalate (subsquare of order 2). • When the intercalate is replaced by the other possible subsquare on the same symbols, the resulting Latin square is in the same species as Ln. Hence Ln generalises the square that Sade famously found to complete Norton’s enumeration of Latin squares of order 7. In particular, Ln is what is known as a self-switching Latin square and possesses a near-autoparatopism.