Half-isomorphisms of moufang loops of odd order

摘要

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 W. R. Scott [Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 11411144] that if G is a group then φ is either an isomorphism or an anti-isomorphism. This is used to prove that a finite group is determined by its group determinant. Here we show that every half-isomorphism between Moufang loops of odd order is either an isomorphism or an anti-isomorphism. Such a result does not carry over to Moufang loops of even order.