Every Moufang loop of odd order has nontrivial nucleus

Abstract

Glauberman [10] and Doro [8] studied the structure of Moufang loops of odd order. Glauberman [10] proved that Feit–Thompson's Theorem can be extended to Moufang loops, namely every Moufang loop of odd order is solvable. It turned out that the multiplication group G of a Moufang loop of odd order with trivial nucleus is a group with triality, i.e. S3≤AutG with special identities.One of the main problems in Moufang loops: Does there exist a Moufang loop of odd order with trivial nucleus? We give a negative answer by proving that every Moufang loop of odd order has nontrivial nucleus.