On half-automorphisms of certain Moufang loops with even order

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. It was shown by Scott (1957) [8], that if G is a group then φ is either an isomorphism or an anti-isomorphism. This, along with Frobeniusʼ original papers on character theory, was used to prove that a finite group is determined by its group determinant. It was then shown by Gagola and Giuliani (2012) [5] that Scottʼs result carries over to Moufang loops of odd order. However, such a result does not hold for all Moufang loops that are of even order. Here we look at certain Moufang loops of even order and determine under what conditions is a half-automorphism forced to be either an automorphism or an anti-automorphism.