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

Schmid, Wolfgang A.

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 323-345.

Soit H un monoïde de Krull dont le groupe de classes est fini. On suppose que chaque classe contient un diviseur premier. On sait que tout ensemble de longueurs est une presque multiprogression arithmétique. Nous étudions les nombres entiers qui apparaissent comme raison de ces progressions. Nous obtenons en particulier une borne supérieure sur la taille de ces raisons. En appliquant ces résultats, nous pouvons montrer que, sauf dans un cas particulier connu, deux p-groupes élémentaires ont le même système d’ensembles de longueurs si et seulement si ils sont isomorphes.

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.

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