Structural patterns of autotopisms of maximum rank quasigroups

Abstract

In this paper, some properties of the set Qn of those quasigroups of n elements having maximum rank n are studied. Although one such a quasigroup Q must be a loop, the reciprocal is false in general. So, the existence of an unit element of Q can be used in order to study the symmetrical structure of its multiplication table, given by the autotopism group of Q. Moreover, by imposing the condition of having maximum rank, a classification of all possible structural patterns of Qn can be obtained. Finally, it is given an outline about the application of all the previous results in the calculus of the character tables of the quasigroups of Qn and their corresponding determinant groups.