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.
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 
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/
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     author = {Aboras, Mouna},
     title = {Dihedral-like constructions of automorphic loops},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {269--284},
     year = {2014},
     volume = {55},
     number = {3},
     mrnumber = {3225610},
     zbl = {06391543},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2014_55_3_a1/}
}
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