Hajós' theorem for multiple factorizations
type: Item
instance of: scholarly article
title: Hajós' theorem for multiple factorizations
zbMATH Open document ID: 0821.20032
DOI: 10.1007/BF01874256
published in: Acta Mathematica Hungarica
publication date: 1995-09-28T00:00:00Z
review text: Let \(G\) be a finite abelian group written multiplicatively with identity element \(e\). Let \(A_ 1, \dots, A_ n\) be subsets of \(G\). If each element \(g\) of \(G\) is expressible in precisely \(k\) ways in the form \(g = a_ 1 \dots a_ n\), \(a_ 1 \in A_ 1, \dots, a_ n \in A_ n\), then we say that the product \(A_ 1 \dots A_ n\) is a \(k\)-factorization of \(G\). When the product \(A_ 1 \dots A_ n\) is direct than it is a 1- factorization of \(G\) and will be called simply a factorization of \(G\). The subset \(A\) of \(G\) is defined to be cyclic if it is of the form \(\{e, a, a^ 2, \dots, a^{r-1}\}\). \textit{G. Hajós} proved that in every factorization of a finite abelian group by cyclic subsets at least one subgroup must occur among the factors [Math. Z. 47, 427-467 (1941; Zbl 0025.25401)]. The authors first give examples which show that Hajós' theorem does not extend to multiple factorizations, and then prove the following theorem: In a multiple factorization of a finite abelian group by cyclic subsets a subgroup always occurs among the factors, if the multiplicity is relatively prime to the order of the group.
reviewed by: Yongcai Ren
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 638233
zbMATH Keywords: factorization by cyclic subsets
zbMATH Keywords: finite Abelian groups
zbMATH Keywords: subgroups
zbMATH Keywords: Hajós theorem
zbMATH Keywords: multiple factorizations
author: K. Corrádi
author: Sándor Szabó
MaRDI profile type: Publication
Wikidata QID: Q114233912
cites work: Über einfache und mehrfache Bedeckung des \(n\)-dimensionalen Raumes mit einem Würfelgitter
cites work: Die neue Theorie der endlichen abelschen Gruppen und Verallgemeinerung des Hauptsatzes von Hajós
full work available at URL: https://doi.org/10.1007/bf01874256
OpenAlex ID: W2087235586
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description: scientific article; zbMATH DE number 638233