Lie automorphic loops under half-automorphisms

摘要

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 (ab) is an element of {f (a) f (b), f (b) f (a)} for any a, b is an element of 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.