Half-isomorphisms of dihedral automorphic loops. Issue 3 (3rd March 2020) - Giuliani, Maria de Lourdes Merlini; dos Anjos, Giliard Souza
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 D i h ( α, G ) . 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 2 | G | . A half-isomorphism of loops is a bijection f between loops L, L ′ where for any x, y ∈ L, we have f ( x y ) ∈ f ( x ) f ( y ), f ( y ) f ( x ) . We say that a half-isomorphism is nontrivial when it is neither an isomorphism nor an anti-isomorphism. In this paper, we prove that D i h ( α, G ) 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.