On Some Autotopisms Of Non-Steiner Central Loops

TL;DR

This paper introduces a method to construct autotopisms in non-Steiner C-loops, demonstrating how certain autotopisms relate to the equivalence of parastrophes and characterizing their structure as a Steiner triple system.

Contribution

It provides a novel algebraic process for constructing autotopisms in non-Steiner C-loops and links these autotopisms to the structure of Steiner triple systems.

Findings

Constructed autotopisms prevent C-loops from being Steiner loops.

Two parastrophes are not equivalent iff specific autotopisms are non-identity.

Autotopisms form a Steiner triple system.

Abstract

An algebraic process for the construction of an autotopism for a non-Steiner C-loop is described and this is demonstrated with an example using a known finite C-loop. In every C-loop, two of its parastrophes are not equivalent(equal) it, if and only if both the first and second components of the constructed autotopism and its inverse autotopism are not equal to the identity map. Hence, none of the other three parastrophes is equivalent(equal) to the C-loop. It is proved that the set of autotopisms that prevent a C-loop from being a Steiner loop forms a Steiner triple system.