A characterization of class groups via sets of lengths II
Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 327-346.

Soit H un monoïde de Krull avec un groupe des classes fini G, et supposons que chaque classe contient un diviseur premier. Si un élément aH a une factorisation a=u 1 u k en éléments irréductibles u 1 ,...,u k H, alors nous appelons k la longueur de la factorisation et l’ensemble L(a) de toutes les longueurs de factorisation possibles l’ensemble des longueurs de a. C’est bien connu que le système (H)={L(a)aH} de tous les ensembles de longueurs ne dépend que du groupe des classes G, et c’est bien une conjecture de longue date que, inversement, le système (H) caractérise le groupe des classes. Nous vérifions la conjecture si le groupe des classes est isomorphe à C n r avec r,n2 et rmax{2,(n+2)/6}.

En effet, soit H ' un autre monoïde de Krull avec un groupe des classes G ' tel que chaque classe contient un diviseur premier, et supposons que (H)=(H ' ). Nous montrons que, si l’un des groupes G et G ' est isomorphe à C n r avec r,n donnés comme ci-dessus, alors G et G ' sont isomorphes (à part deux exceptions bien connues).

Let H be a Krull monoid with finite class group G and suppose that every class contains a prime divisor. If an element aH has a factorization a=u 1 ·...·u k into irreducible elements u 1 ,...,u k H, then k is called the length of the factorization and the set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system (H)={L(a)aH} of all sets of lengths depends only on the class group G, and a standing conjecture states that conversely the system (H) is characteristic for the class group. We verify the conjecture if the class group is isomorphic to C n r with r,n2 and rmax{2,(n+2)/6}. Indeed, let H ' be a further Krull monoid with class group G ' such that every class contains a prime divisor and suppose that (H)=(H ' ). We prove that, if one of the groups G and G ' is isomorphic to C n r with r,n as above, then G and G ' are isomorphic (apart from two well-known pairings).

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DOI : https://doi.org/10.5802/jtnb.983
Classification : 11B30,  11R27,  13A05,  13F05,  20M13
Mots clés : Krull monoids, maximal orders, seminormal orders, class groups, arithmetical characterizations, sets of lengths, zero-sum sequences, Davenport constant
@article{JTNB_2017__29_2_327_0,
     author = {Alfred Geroldinger and Qinghai Zhong},
     title = {A characterization of class groups via sets of lengths {II}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {327--346},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {2},
     year = {2017},
     doi = {10.5802/jtnb.983},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.983/}
}
Alfred Geroldinger; Qinghai Zhong. A characterization of class groups via sets of lengths II. Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 327-346. doi : 10.5802/jtnb.983. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.983/

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