On the factorisation of finite abelian groups. III

Abstract

A famous conjecture of Minkowski, concerning the columnation of space-filling lattices, was first proved by Hajos in 1941 by translating the problem into one involving finite abelian groups. The problem solved by Hajos was one concerning a special type of factorisation of finite abelian groups. In the general problem considered in the thesis no restriction is placed on the nature of the factors. It was originally conjectured by Hajos that in any factorisation; one of the factors must possess a non-trivial subgroup as a factor, However, Hajos himself soon found that not all finite abelian groups possess this property. Those which do were called "good" and those which do not were called "bad" Further contributions to determining those groups which are good and those which are had were made by Redei and de Bruijn. But for groups of many types the problem was left undecided. In this thesis the problem is solved completely for finite abelian groups. A special case of this problem for cyclic groups was shown by de Bruijn to be equivalent to a conjecture of his concerning bases for the sets of integers. This conjecture and a generalisation of it are also shown to be true. It is shown first that a cyclotomic polynomial is irreducible over certain fields of reots of unity. This extension of the well-known result that a eyclotomic polynomial is irreducible over the rational field is basic to the following work and is used frecuently throughout the thesis. A theorem, similer to the theorems of de Bruijn, showing that certain types of groups are had is then proved, then, in the main part of the thesis all the groups not shown to be bad by this theorem or one of the theorems of de Bruijn are shown to be good. Hajos gave a method which, be claimed, would give all factorisations of a good group. However it is shown that a correction is needed in this method and the corrected method is then presented. The final section is concerned with the extension of the results to certain types of infinite abelian groups. Under the restriction that one of the factors shall have only a finite number of elements, similar results to these proved for finite groups are obtained for the generalisations of these groups to the infinite cases.