Lie automorphic loops under half-automorphisms.

Automorphic loops or A -loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring (Q , + , [. ,. ]) we can define an operation (∗) such that (Q , ∗) is an A -loop. We call it Lie automorphic loop. A half-isomorphism f : G → K between multiplicative systems G and K is a bijection from G onto K such that f (a b) ∈ { f (a) f (b) , f (b) f (a) } for any a , b ∈ G. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc.8 (1957) 1141–1144] that if G is a group then f 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. [ABSTRACT FROM AUTHOR]