On Some Autotopisms Of Non-Steiner Central Loops

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.