Every Moufang loop of odd order has nontrivial nucleus (Q2134017)

In this paper, the author give a negative answer by proving that every Moufang loop of odd order has nontrivial nucleus. To obtain his result the author focus on Moufang loops satisfying any one of the three equivalent Moufang identities: $((xy)x)z = x(y(xz))$; $((xy)z)y = x(y(zy))$; $(xy)(zx) = x(yz)x$. A class of Moufang loop can be viewed as a ``group with weakened associativity''. The links between algebra, geometry, and group theory explain the importance of this class. Thus, according to the author, the main objective of the paper is to solve one of the main problems in the area of Moufang loops, namely: does there exist a Moufang loop of odd order with trivial nucleus? So, the author gives a negative answer. For the proof of the main result, the author used the structural properties of the multiplication group of Moufang loops of odd order. The author's proof is completely group theoretical relying on the theory of connected transversals. This concept was introduced by \textit{M. Niemenmaa} and \textit{T. Kepka} [J. Algebra 135, No. 1, 112--122 (1990; Zbl 0706.20046)]. Using their characterization theorem, the author transforms loop theoretical problems into group theoretical problems.