Odd order semidirect extensions of commutative automorphic loops

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Displaying similar documents to “Odd order semidirect extensions of commutative automorphic loops”

Dihedral-like constructions of automorphic loops

Mouna Aboras (2014)

Commentationes Mathematicae Universitatis Carolinae

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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 $α \in Aut (G)$ , let $Dih (m, G, α)$ be defined on $ℤ_{m} \times G$ by $(i, u) (j, v) = (i \oplus j, ({(- 1)}^{j} u + v) α^{i j}) .$ The resulting loop is automorphic if and only if $m = 2$ or ( $α^{2} = 1$ 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.

On centerless commutative automorphic loops

Gábor P. Nagy (2014)

Commentationes Mathematicae Universitatis Carolinae

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In this short paper, we survey the results on commutative automorphic loops and give a new construction method. Using this method, we present new classes of commutative automorphic loops of exponent $2$ with trivial center.

Schreier loops

Péter T. Nagy, Karl Strambach (2008)

Czechoslovak Mathematical Journal

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We study systematically the natural generalization of Schreier's extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes.

Loops which are cyclic extensions of their nuclei

Edgar G. Goodaire, D. A. Robinson (1982)

Compositio Mathematica

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Possible orders of nonassociative Moufang loops

Orin Chein, Andrew Rajah (2000)

Commentationes Mathematicae Universitatis Carolinae

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The paper surveys the known results concerning the question: “For what values of $n$ does there exist a nonassociative Moufang loop of order $n$ ?” Proofs of the newest results for $n$ odd, and a complete resolution of the case $n$ even are also presented.