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 ≥ 1 and α ∈ Aut ( G ) , let Dih ( m , G , α ) be defined on ℤ m × G by ( i , u ) ( j , v ) = ( i ⊕ 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.