Automorphisms of dihedral-like automorphic loops

Abstract

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 $\alpha$ an automorphism of $G$ that satisfies ${\alpha }^2=1$ if $m>2$. Then the dihedral-like automorphic loop $\mathrm{Dih}(m,G,\alpha )$ is defined on ${\mathbb{Z}}_m\times G$ by $(i,u)(j,v)=(i+j,((-1{)}^{j}u+v){\alpha }^{ij})$. We prove that two finite dihedral-like automorphic loops $\mathrm{Dih}(m,G,\alpha )$, $\mathrm{Dih}(\overline{m},\overline{G},\overline{\alpha })$ are isomorphic if and only if $m=\overline{m}$, $G=\overline{G}$, and $\alpha$ is conjugate to $\overline{\alpha }$ 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 $Q$.