A GENERALIZATION OF HAJOS’ THEOREM

Year 2008, Volume: 3 Issue: 3, 83 - 95, 01.06.2008

A. D. Sands

Abstract

Haj´os’ Theorem states that if a finite abelian group is expressed as a direct product of cyclic subsets, then one of these subsets must be a subgroup. Here factorizations are considered in which one of the factors is not assumed to be cyclic but has certain restrictions on its order placed upon it.

Keywords

factorization, abelian group, cyclic subset

References

• N.G. Brujin, On the factorization of finite abelian groups, Indag. Math. 15 (1953), 258–264.
• K. Corr´adi, A.D. Sands and S. Szab´o, Factoring by simulated subsets, J. Alge- bra, 175 (1995), 320–331.
• K. Corr´adi, A.D. Sands and S. Szab´o, Factoring by simulated subsets II, Comm. Algebra, 27 (1999), 5367–5376.
• G. Haj´os, ¨Uber einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einem W¨urfelgitter, Math. Zeitschrift, 47 (1942), 427–467.
• L. R´edei, Die neue Theorie der endlichen Abelschen Gruppen und Verallge- meinerung des Hauptsatzes von Haj´os, Acta Math. Hungar. 16 (1965), 329– 373.
• A.D. Sands, Factorization of cyclic groups, Proc. Coll. Abelian Groups (Ti- hany), (1964), 139–146.
• A.D. Sands, Factorizations of abelian groups involving simulated factors and one other factor, Acta Sci. Math. to appear.
• A.D. Sands and S. Szab´o, Factorization of periodic subsets, Acta Math. Hun- gar. 57 (1991), 159–167.
• S. Szab´o, Factoring by cyclic and simulated subsets, Acta Math. Hungar. 85 (1999), 123–133. A.D. Sands
• Department of Mathematics Dundee University
• Dundee DD1 4HN Scotland
• E-mail: [e-mail]