On large half-factorial sets in elementary \(p\)-groups: maximal cardinality and structural char\-ac\-ter\-i\-zation.
type: Item
instance of: scholarly article
title: On large half-factorial sets in elementary \(p\)-groups: maximal cardinality and structural char\-ac\-ter\-i\-zation.
zbMATH Open document ID: 1072.20032
DOI: 10.1007/BF02786695
author: Alain Plagne
author: Wolfgang A. Schmid
published in: Israel Journal of Mathematics
publication date: 2005-05-03T00:00:00Z
review text: With a subset \(G_0\) of a finite Abelian group \(G\) (written additively) one associates the free monoid \({\mathcal F}(G_0)\) whose free generators are elements of \(G_0\), and the block monoid \({\mathcal B}(G_0)\), consisting of all \(\prod_{i=1}^mg_i\in{\mathcal F}(G_0)\) with \(\sum_{i=1}^mg_i=0\). \(G_0\) is called half-factorial, if \({\mathcal B}(G_0)\) is half-factorial. The authors obtain an explicit formula for \(\mu(G)\), the maximal cardinality of a half-factorial subset of \(G\) in the case when \(G=C_p^n\). This extends a result of \textit{A. Geroldinger} and \textit{J. Ka\-czo\-rowski} [Sémin. Théor. Nombres, Bordx., Sér. II 4, No. 2, 199-238 (1992; Zbl 0780.11046)], who obtained this formula in the case of even \(n\). Moreover, the authors determine the structure of half-factorial subsets \(G_0\) of \(C_p^n\) having at least \(\mu(G)-cp\) elements, with \(c\) being an absolute constant. These results are then applied to the problem of oscillations of the error term in the counting formula for the set of non-associated integers in an algebraic number field (with class-group \(C_p^n\)) having factorizations of at most \(k\) different lengths.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 11R47
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 20M14
zbMATH DE Number: 2164545
zbMATH Keywords: half-factorial sets
zbMATH Keywords: factorizations
zbMATH Keywords: finite Abelian groups
MaRDI profile type: Publication
cites work: Q4342479
cites work: Q4297849
cites work: Q4320565
cites work: A Characterization of Algebraic Number Fields with Class Number Two
cites work: Half-factorial domains, a survey
cites work: Q4343533
cites work: Factorization in Dedekind domains with finite class group
cites work: Every Abelian group is a class group
cites work: Q5670698
cites work: Half-factorial domains and half-factorial subsets of Abelian groups
cites work: Systems of sets of lengths. II
cites work: On non-unique factorizations into irreducible elements
cites work: Ein quantitatives Resultat über Faktorisierungen verschiedener Länge in algebraischen Zahlkörpern. (A quantitative result on factorizations of different length in algebraic number fields)
cites work: Systems of sets of lengths.
cites work: Q4679048
cites work: Q4320501
cites work: Analytic and arithmetic theory of semigroups with divisor theory
cites work: Halbgruppen mit Divisorentheorie. (Semigroups with divisor theory)
cites work: Q4343532
cites work: Q4255424
cites work: Q4811791
cites work: Some remarks on factorization in algebraic number fields
cites work: Q5465426
cites work: Q1082376
cites work: On algebraic number fields with non-unique factorization
cites work: Finite abelian groups and factorization problems
cites work: Oscillations of error terms associated with certain arithmetical functions
cites work: On the distribution of algebraic numbers with prescribed factorization properties
cites work: On the asymptotic behavior of some counting functions
cites work: Q3410836
cites work: On the asymptotic behavior of some counting functions, II
cites work: On c-semigroups
cites work: Factorizations of distinct lengths in algebraic number fields
cites work: Remarks on factorizations in algebraic number fields
cites work: Half factorial domains
cites work: Half-factorial-domains
Recommended article: On the maximal cardinality of half-factorial sets in cyclic groups
Recommended article: On the asymptotic behavior of some counting functions
Recommended article: Q3410836
Recommended article: Q3430012
Recommended article: Half-factorial domains and half-factorial subsets of Abelian groups
Recommended article: EZADS inputs which produce half-factorial block monoids.
Recommended article: Q4811791
Recommended article: On congruence half-factorial Krull monoids with cyclic class group
Recommended article: Weakly half-factorial sets in finite abelian groups
Recommended article: Q3621981
description: scientific article; zbMATH DE number 2164545