SAC loops and loops of order five

Abstract

In this article, we continue the analytical research of small-order loops. Namely, we investigate loops of order 5. Recall that an element of a loop is called unipotent if its square is neutral. A loop is called unipotent if all its elements are unipotent.
One of loops of order 5 is a semisymmetric anticommutative loop (SAC loop). The following property is true: ``If a unipotent loop is isotopic to an SAC loop, then the components of the isotopy coincide, so the loops are isomorphic.'' Since any SAC loop is unipotent, any isotopism (autotopism) is an isomorphism (respectively, an automorphism) in the class of SAC loops. This property allowed us to describe the isomorphism relation on the isotopes of the SAC loop. As a consequence, we obtain a complete classification of loops of order 5 and each of their automorphism group. In addition, we managed to solve the recognition problem for all six loops of order 5. For example, a 5-order loop is isomorphic to: 1) the group if and only if the squares of all elements are pairwise different; 2) SAC loop if and only if it has at least three unipotents.