On centerless commutative automorphic loops

Gábor P. Nagy

Displaying similar documents to “On centerless 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.

Odd order semidirect extensions of commutative automorphic loops

Přemysl Jedlička (2014)

Commentationes Mathematicae Universitatis Carolinae

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We analyze semidirect extensions of middle nuclei of commutative automorphic loops. We find a less complicated conditions for the semidirect construction when the middle nucleus is an odd order abelian group. We then use the description to study extensions of orders $3$ and $5$ .

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.