Half-isomorfismos de loops automórficos diedrais

Abstract

Automorphic loops, or A-loops, are loops in which all inner mappings are automorphisms. The dihedral automorphic loops class considered in this thesis are constructed by taking the cyclic groups Zm and Cp where m is an even positive integer and p is a prime number greater than or equal to 2. We denote such loops by La = Dih(m, Cp, a), where a is an automorphism of Cp. A half-isomorphism between La and Lb is a bijection f such that f (x y) = f (x) f (y) or f (x y) = f (y) f (x), for any elements x, y 2 La. We say that f is nontrivial if f is neither isomorphism nor anti-isomorphism. In this thesis we describe the properties and classify all nontrivial half-isomorphisms between La and Lb. We also prove that in LJ all half-automorphisms are trivial where J is the automorphism of Cp defined by J(x) = x􀀀1.