Gröbner bases and the number of Latin squares related to autotopisms of order ≤7

Abstract

Latin squares can be seen as multiplication tables of quasigroups, which are, in general, noncommutativeand non-associative algebraic structures. The number of Latin squares having a fixed isotopismin their autotopism group is at the moment an open problem. In this paper, we use Gröbner bases todescribe an algorithm that allows one to obtain the previous number. Specifically, this algorithm is implemented in SINGULAR to obtain the number of Latin squares related to any autotopism of Latin squares oforder up to 7.