Congruence Solvability in Finite Moufang Loops of Order Coprime to Three

We prove that a normal subloop X of a Moufang loop Q induces an abelian congruence of Q if and only if u(xy)=(uy)x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(xy) = (uy)x$$\end{document} for all x,y is an element of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\in X$$\end{document} and u is an element of Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in Q$$\end{document}. This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable.