On the nuclei of Moufang loops with orders coprime to six

An open problem, originally proposed by J.D. Phillips, asks if there exists an odd ordered Moufang loop that possesses a trivial nucleus. In 1968 George Glauberman proved [7] that if Q is a Moufang loop of odd order and M is any minimal normal subloop of Q whose order is coprime to its index in Q, then M is contained in the nucleus of Q. We are able to strengthen Glaubermanʼs result here by removing the coprime assumption between the order of M and its index in Q given that the loop Q has an order not divisible by three (in addition to being of odd order). Thus, a nontrivial Moufang loop having an order coprime to six certainly has a nontrivial nucleus. Concerning then the question raised by J.D. Phillips, any nontrivial Moufang loop of odd order with a trivial nucleus (should one exist) must have an order divisible by three.