Classification of group isotopes according to their symmetry groups

TL;DR

This paper classifies group isotopes based on their symmetry groups, providing criteria for their categorization into various quasigroup classes and applying these results to classify linear group isotopes of prime orders.

Contribution

It introduces criteria for classifying group isotopes into symmetry-based classes and applies these to classify linear group isotopes of prime orders.

Findings

Noncommutative group isotopes are either semi-symmetric or asymmetric.

Non-medial T-quasigroups are asymmetric.

Classification of linear group isotopes of prime orders.

Abstract

The class of all quasigroups is covered by six classes: the class of all asymmetric quasigroups and five varieties of quasigroups (commutative, left symmetric, right symmetric, semi-symmetric and totally symmetric). Each of these classes is characterized by symmetry groups of its quasigroups. In this article, criteria of belonging of group isotopes to each of these classes are found, including the corollaries for linear, medial and central quasigroups etc. It is established that an isotope of a noncommutative group is either semi-symmetric or asymmetric, each non-medial T-quasigroup is asymmetric etc. The obtained results are applied for the classification of linear group isotopes of prime orders, taking into account their up to isomorphism description.