Lie automorphic loops under half-automorphisms

Abstract

Automorphic loops or [Formula: see text]-loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring [Formula: see text] we can define an operation [Formula: see text] such that [Formula: see text] is an [Formula: see text]-loop. We call it Lie automorphic loop. A half-isomorphism [Formula: see text] between multiplicative systems [Formula: see text] and [Formula: see text] is a bijection from [Formula: see text] onto [Formula: see text] such that [Formula: see text] for any [Formula: see text]. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 1141–1144] that if [Formula: see text] is a group then [Formula: see text] is either an isomorphism or an anti-isomorphism. This was used to prove that a finite group is determined by its group determinant. Here, we show that every half-automorphism of a Lie automorphic loop of odd order is either an automorphism or an anti-automorphism.