A short proof for the central nilpotency of Moufang loops of prime power order

Abstract

The main result is an elementary proof of the nilpotency of Moufang loops which are of prime power order. Besides basic properties of Moufang loops and Sylow theorems, the proof relies on the fact that a relative multiplication group of a (centrally nilpotent) subloop of order pk is a p-group in any finite Moufang loop Q, and that Lφ(x)−1φLxφ−1 and Rφ(x)−1φRxφ−1 are pseudoautomorphisms of Q whenever x∈Q and φ is a pseudoautomorphism of Q.