An extension for Hajós' theorem
type: Item
instance of: scholarly article
title: An extension for Hajós' theorem
zbMATH Open document ID: 0798.20047
DOI: 10.1016/0022-4049(92)90051-G
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: \textit{G. Hajós}' theorem asserts that if a finite abelian group is expressed as a direct product of cyclic subsets of prime cardinality, then at least one of the factors must be a subgroup [see Math. Z. 47, 427-467 (1941; Zbl 0025.25401)]. \textit{A. D. Sands} proved that in a factorization of a finite cyclic group by subsets of cardinalities that are prime powers at least one of the factors is periodic. [Proc. Colloq. Abelian Groups, 139-146 (1964; Zbl 0132.271)]. In the reviewed paper the authors prove the following Theorem 5: Let \(G\) be a finite abelian group and let \(R\) be a subset of the prime divisors of \(|G|\). Suppose that the \(r\)-components of \(G\) are cyclic for each \(r \in R\). If \(G = A_ 1\cdots A_ n\) is a normed factorization of \(G\), where for each \(i\), \(1 \leq i\leq n\), either \(A_ i\) is a pseudocyclic subset and hence of prime order or \(|A_i|\) is a power of one of the primes \(r \in R\), then one of the factors \(A_ i\) is periodic. The authors also show that pseudocyclicity is a generalization of cyclicity, and that if a normed subset of prime cardinality of \(G\) is periodic, then it is a subgroup. Hence, the theorem is a common generalization of Hajós' and Sands' theorem.
reviewed by: Yongcai Ren
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 05B40
zbMATH DE Number: 61224
zbMATH Keywords: finite abelian group
zbMATH Keywords: direct product of cyclic subsets
zbMATH Keywords: finite cyclic group
zbMATH Keywords: periodic
zbMATH Keywords: normed factorization
zbMATH Keywords: pseudocyclic subset
zbMATH Keywords: normed subset
author: K. Corrádi
author: Sándor Szabó
MaRDI profile type: Publication
cites work: A new proof of Rédei's theorem
cites work: Solution to a problem of A. D. Sands
cites work: Die neue Theorie der endlichen abelschen Gruppen und Verallgemeinerung des Hauptsatzes von Hajós
cites work: Q5341835
full work available at URL: 0022-4049(92)90051-g
OpenAlex ID: W2034819683
description: scientific article; zbMATH DE number 61224