Automorphisms of dihedral-like automorphic loops

TL;DR

This paper characterizes the isomorphism classes of dihedral-like automorphic loops constructed from abelian groups and automorphisms, and describes their automorphism groups and inner mappings.

Contribution

It provides a complete classification of finite dihedral-like automorphic loops and details their automorphism and inner mapping groups.

Findings

Two such loops are isomorphic iff their parameters are equal and automorphisms are conjugate.

The structure of automorphism groups of these loops is explicitly described.

Inner mappings form a subgroup with a known structure.

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 $Q$.