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 $Dih$$(\alpha ,G).$ This class is constructed from a finite abelian group G and an automorphism of G, generalizing the construction of the dihedral group which has order $2\left|G\right|.$ A half-isomorphism of loops is a bijection f between loops $L,L\prime$ where for any $x,y\in L,$ we have $f(xy)\in \{f(x)f(y),f(y)f(x)\}.$ We say that a half-isomorphism is nontrivial when it is neither an isomorphism nor an anti-isomorphism. In this paper, we prove that $Dih$$(\alpha ,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.

Keywords:

• Automorphic loops
• dihedral automorphic loops
• half-isomorphism
• loops

Mathematics Subject Classification:

• 20N05

Acknowledgments

Thanks to the referee for useful suggestions.

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - The Open Funder Registry code is 402083.