Half-automorphisms of simple Moufang loops

Abstract

A half-isomorphism φ:G→K between multiplicative systems G and K is a bijection from G onto K such that φ(ab)∈{φ(a)φ(b),φ(b)φ(a)} for any a,b∈G. For groups all half-isomorphisms are either isomorphisms or anti-isomorphisms as shown by W.R. Scott (1957) [14]. Scott's result carries over to certain classes of Moufang loops including Moufang loops of odd order. The class of loops considered here are Paige loops, namely the class of finite nonassociative simple Moufang loops. In this paper, it is shown that any half-automorphism of a finite simple Moufang loop is either an automorphism are an anti-automorphism, thus extending Scott's result.