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On the factorisation of finite abelian groups. III

On the factorisation of finite abelian groups. III Aeta Mathematica Academiae Scientiarum Hungaricae Tomus 25 (3--4), (1974), pp. 279--284. ON THE FACTORISATION OF FINITE ABELIAN GROUPS. III By A. D. SANDS (Dundee) In Memoriam, Gy6rgy Haj6s 1. Introduction In a previous paper [7] it was shown that if in a factorisation of a finite cyclic group G one factor has order a power of a prime p then one of the two factors is periodic. This proved and generalised a conjecture of DE BRUIJN [2]. The main purpose of this paper is to further generalise this theorem by showing that the con- clusion holds whenever G is a finite abelian group with cyclic p-component. Genera- lisations of this to cases involving more than two factors and to certain infinite groups are then deduced. The problem of the factorisation of finite abelian groups arose from the solution by HAJdS [3] of a classical problem of Minkowski. The definitions and notations used in this paper may be found in [7]. In the cyclic case considered there, it was not necessary to use representation theory formally, though the work involving roots of unity is essentially of this type. For the generalisation given here we make use of certain elementary facts http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

On the factorisation of finite abelian groups. III

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01886085
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Abstract

Aeta Mathematica Academiae Scientiarum Hungaricae Tomus 25 (3--4), (1974), pp. 279--284. ON THE FACTORISATION OF FINITE ABELIAN GROUPS. III By A. D. SANDS (Dundee) In Memoriam, Gy6rgy Haj6s 1. Introduction In a previous paper [7] it was shown that if in a factorisation of a finite cyclic group G one factor has order a power of a prime p then one of the two factors is periodic. This proved and generalised a conjecture of DE BRUIJN [2]. The main purpose of this paper is to further generalise this theorem by showing that the con- clusion holds whenever G is a finite abelian group with cyclic p-component. Genera- lisations of this to cases involving more than two factors and to certain infinite groups are then deduced. The problem of the factorisation of finite abelian groups arose from the solution by HAJdS [3] of a classical problem of Minkowski. The definitions and notations used in this paper may be found in [7]. In the cyclic case considered there, it was not necessary to use representation theory formally, though the work involving roots of unity is essentially of this type. For the generalisation given here we make use of certain elementary facts

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Sep 1, 1974

References

  • On the factorization of finite abelian groups
    Bruijn, N. G.
  • On the factorization of cyclic groups
    Bruijn, N. G.
  • Über einfache und mehrfache Bedeckung desn-dimensionalen Raumes mit einem Würfelgitter
    Hajós, G.
  • Sur le problème de factorisation des groupes cycliques
    Hajós, G.
  • Mémoire sur les facteurs irréductibles de l'expressionx n−1
    Kronecker, L.
  • Zwei Lückensatze über Polynome in endlichen Primkörpern mit Anwendungen auf die endlichen Abelschen Gruppen und die Gaussischen Summen
    Rédei, L.
  • On the factorisation of finite abelian groups
    Sands, A. D.
  • The factorisation of abelian groups
    Sands, A. D.
  • On the factorisation of finite abelian groups. II
    Sands, A. D.