On the Derivatives of Central Loops

The right(left) derivative, a⁻¹, e− and e, a⁻¹− isotopes of a C-loop are shown to be C-loops. Furthermore, for a central loop (L, F), it is shown that {F, F^a−1, F_{a⁻¹, e}} and {F, F_a⁻¹, F_e,a⁻¹} are systems of isotopic C-loops that obey a form of generalised distributive law. It is proved that for a loop (L, θ) to be an LC(RC,C)-loop, it is necessary and sufficient for the parastrophe (L, θ*) to be a RC(LC,C)-loop. Hence, isotopes (L, ⊗) and (L, ⊖) of (L, θ) and (L, θ*) respectively are proved to be isotopic if either (L, ⊗) or (L, ⊖) is commutative. It is shown that C-loops are isotopic to some finite indecomposable groups of the classes D_i,i = 1, 2, 3, 4, 5 and that the center of such C-loops have a rank of 1, 2 or 3.