Geodesic
Dihedral-like constructions of automorphic loops
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 3, pp. 269-284

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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.
Classification : 20N05
Keywords: dihedral automorphic loop; automorphic loop; inner mapping group; multiplication group; nucleus; commutant; center; commutator; associator subloop; derived subloop 
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     title = {Dihedral-like constructions of automorphic loops},
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Aboras, Mouna. Dihedral-like constructions of automorphic loops. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 3, pp. 269-284. http://geodesic.mathdoc.fr/item/CMUC_2014__55_3_a1/