Half-isomorphisms of dihedral automorphic loops

Abstract

Automorphic loops, or A-loops, are loops in which all inner mappings are automorphisms. Here, we investigate a class of A-loops known as the dihedral automorphic loop, denoted by This class is constructed from a finite abelian group G and an automorphism of G, generalizing the construction of the dihedral group which has order A half-isomorphism of loops is a bijection f between loops where for any we have We say that a half-isomorphism is nontrivial when it is neither an isomorphism nor an anti-isomorphism. In this paper, we prove that has nontrivial half-isomorphisms and we classify nontrivial half-isomorphisms between dihedral automorphic loops. Also we identify the group of half-automorphisms of this class of A-loops.