The $S_{3}$ symmetry: Flavour and texture zeroes

We use the permutational symmetry group $S_{3}$ as a symmetry of flavour, which leads to a unified treatment of masses and mixings of the quarks and leptons. In this framework all mass matrices of the fermions in the theory have the same form with four texture zeroes of class of I. Also, with the help of six elements of real matrix representation of $S_ {3}$ as transformation matrices of similarity classes, we make a classification of the sets of mass matrices with texture zeroes in equivalence classes. This classification reduce the number of phenomenologically viable textures for the non-singulars mass matrices of $3\times3$, from thirty three down to only eleven independent sets of matrices. Each of these sets of matrices has exactly the same physical content.

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