Half-isomorphisms of dihedral automorphic loops. Issue 3 (3rd March 2020)

• Record Type: Journal Article
• Authors: Giuliani, Maria de Lourdes Merlini; dos Anjos, Giliard Souza
• Is Part Of: Communications in algebra. Volume 48:Issue 3(2020)
• Page Start: 1150
• Page End: 1162
• Publication Date: 2020-03-03
• Subjects: Automorphic loops -- dihedral automorphic loops -- half-isomorphism -- loops; 20N05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2019.1677691
• Languages: English
• ISSNs: 0092-7872

Abstract

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.