Automorphic loops are loops in which all inner mappings are automorphisms. We study a generalization of the dihedral construction for groups. Namely, if $(G,+)$ is an abelian group, $m\ge 1$ and $\alpha \in Aut\left(G\right)$, let $Dih(m,G,\alpha )$ be defined on ${\mathbb{Z}}_m\times G$ by $$(i,u)(j,v)=(i\oplus j,({(-1)}^{j}u+v){\alpha }^{ij}).$$ The resulting loop is automorphic if and only if $m=2$ or (${\alpha }^2=1$ and $m$ is even). The case $m=2$ was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We present several structural results about the automorphic dihedral loops in both cases.

This note contains Sylow's theorem, Lagrange's theorem and Hall's theorem for finite Bruck loops. Moreover, we explore the subloop structure of finite Bruck loops.