Simple right conjugacy closed loops

TL;DR

This paper presents a new general method for constructing finite simple right conjugacy closed loops using linear algebra over finite fields, providing a complete classification and count of such loops.

Contribution

It introduces the first general construction for finite simple right conjugacy closed loops and classifies their isomorphism classes.

Findings

Constructed new classes of simple right conjugacy closed loops

Provided an exact count of non-isomorphic loops for each prime power q

Established conditions under which the loops are simple

Abstract

We give a general construction for right conjugacy closed loops, using $GL(2,q)$ for $q$ a prime power. Under certain conditions, the loops constructed are simple, giving the first general construction for finite, simple right conjugacy closed loops. We give a complete description of the isomorphism classes for the construction, yielding an exact count of non isomorphic loops for each $q$.