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 $Q$.