A GENERALIZATION OF HAJOS’ THEOREM

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

1. N.G. Brujin, On the factorization of finite abelian groups, Indag. Math. 15 (1953), 258–264.
2. K. Corr´adi, A.D. Sands and S. Szab´o, Factoring by simulated subsets, J. Alge- bra, 175 (1995), 320–331.
3. K. Corr´adi, A.D. Sands and S. Szab´o, Factoring by simulated subsets II, Comm. Algebra, 27 (1999), 5367–5376.
4. G. Haj´os, ¨Uber einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einem W¨urfelgitter, Math. Zeitschrift, 47 (1942), 427–467.
5. 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.
6. A.D. Sands, Factorization of cyclic groups, Proc. Coll. Abelian Groups (Ti- hany), (1964), 139–146.
7. A.D. Sands, Factorizations of abelian groups involving simulated factors and one other factor, Acta Sci. Math. to appear.
8. A.D. Sands and S. Szab´o, Factorization of periodic subsets, Acta Math. Hun- gar. 57 (1991), 159–167.

9. S. Szab´o, Factoring by cyclic and simulated subsets, Acta Math. Hungar. 85 (1999), 123–133. A.D. Sands
10. Department of Mathematics Dundee University
11. Dundee DD1 4HN Scotland
12. E-mail: [e-mail]