A characterization of nilpotent Moufang loops of odd order

Glauberman and Wright in [G. G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968) 415-417] proved that a nilpotent Moufang loop is the direct product of p-loops for some primes p, consequently the elements of coprime order commute in a nilpotent Moufang loop. In this paper, we prove that in Moufang loops of odd order this condition is equivalent to the central nilpotence.