summary: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, $m\geq 1$ and $\alpha \in \operatorname{Aut}(G)$, let $\operatorname{Dih} (m,G,\alpha )$ be defined on $\mathbb Z_m\times G$ by \begin{equation*} (i,u)(j,v) = (i\oplus j,\,((-1)^{j}u + v)\alpha^{ij}). \end{equation*} 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.
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机译：摘要：自构循环是其中所有内部映射都是自构的循环。我们研究了群体二面体构造的一般化。即，如果$（G，+）$是阿贝尔群，$ m \ geq 1 $和$ \ alpha \在\ operatorname {Aut}（G）$中，则让$ \ operatorname {Dih}（m，G，\ alpha）$由\ begin {equation *}（i，u）（j，v）=（i \ oplus j，\，（（-1）^ {j} u）在$ \ mathbb Z_m \ times G $上定义+ v）\ alpha ^ {ij}）。 \ end {equation *}当且仅当$ m = 2 $或（$ \ alpha ^ 2 = 1 $并且$ m $是偶数）时，结果循环才是自同构的。案例$ m = 2 $由Kinyon，Kunen，Phillips和Vojtěchovský提出。在这两种情况下，我们介绍了有关自构二面环的几个结构结果。
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