On Centrally and Nuclearly Nilpotent Moufang Loops

Let Q be a finite Moufang loop with nucleus N(Q) and associator subloop 𝒜(Q). First we prove if the factor loop over the nucleus Q/N has nontrivial center, then the center of Q is nontrivial too. By using this result we prove that the centrally nilpotence of Q/N(Q) implies the centrally nilpotence of 𝒜(Q), and we show that, for the centrally nilpotence of a finite Moufang loop, the centrally nilpotence of Q/N(Q) and Q/𝒜(Q) is a necessary and sufficient condition. Finally, as a corollary we give a necessary and sufficient condition for the equivalence of centrally and nuclearly nilpotence of finite Moufang loops, namely, the centrally nilpotence of Q/𝒜(Q).