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Automorphisms of Dihedral-Like Automorphic Loops

Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let m be a positive even integer, G an abelian group, and α an automorphism of G that satisfies α2 = 1 if m > 2. Then the dihedral-like automorphic loop Dih(m, G, α) is defined on ℤm × G by We prove that two finite dihedral-like automorphic loops Dih(m, G, α), are isomorphic if and only if , , and α is conjugate to in the automorphism group of G. Moreover, for a finite dihedral-like automorphic loop Q, we describe the structure of the automorphism group of Q and its subgroup consisting of inner mappings of Q.