Multiplication groups of commutative automorphic p-loops of odd order are p-groups

An automorphic loop (or A-loop) is a loop whose inner mappings are automorphisms. It was recently proved (see P. Jedlička, M.K. Kinyon, P. Vojtěchovský (2012) [8]): A finite commutative A-loop of order a power of an odd prime is centrally nilpotent. In this paper we prove without using the nilpotence that the multiplication group of a finite commutative A-loop of order a power of an odd prime is a p-group, which implies the central nilpotence of such loop, too.