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Differences in sets of lengths of Krull monoids with finite class group

Wolfgang A. Schmid (2005)

Journal de Théorie des Nombres de Bordeaux

Let H be a Krull monoid with finite class group where every class contains some prime divisor. It is known that every set of lengths is an almost arithmetical multiprogression. We investigate which integers occur as differences of these progressions. In particular, we obtain upper bounds for the size of these differences. Then, we apply these results to show that, apart from one known exception, two elementary p -groups have the same system of sets of lengths if and only if they are isomorphic.

Dihedral-like constructions of automorphic loops

Mouna Aboras (2014)

Commentationes Mathematicae Universitatis Carolinae

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 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.

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