The dimension subgroup problem

Abstract

Whether the terms of the lower central series of a group are associated with the powers of the augmentation ideal of its integral group ring is a question which has been open for over thirty years. This question, the dimension subgroup problem, has achieved notoriety for the number of false proofs it has elicited. In this paper, we record general observations on the problem, verify the conjecture for several new classes of groups and develop generalizations which indicate new lines of research. In a subsequent paper [20], we examine the subgroups attached to group rings over arbitrary coefficient domains. The general properties of dimension subgroups are set forth in the first section. Several equivalent formulations of the conjecture are given. The subgroups are calculated in a number of special cases. A minimal counterexample is a finite p-group with cyclic center. In the second section, we take the approach that such reductions make available. The conjecture is established for Abelian-by-cyclic groups and for split extensions of Abelian groups by groups satisfying the conjecture; it follows inductively that the Sylow subgroups of the symmetric groups satisfy the problem and so every p-group embeds in a p-group for which the conjecture is verified. This analysis provides an unexpected bridge to the work of Passi who conjectured a strong result about the properties of factor sets of extensions of divisible groups byp-groups and showed how it would establish the dimension subgroup conjecture. Here another way of using Passi’s conjecture to solve the problem is introduced and applied in a minimal counterexample to obtain new results. The third section places the problem in a wider context: I f G, is associated with I”, then G, is associated with even larger ideals. It may be that these ideals will have to be identified before the dimension subgroup problem can be resolved affirmatively. Some such ideals are discovered by expanding the methods of Passi and Moran. The results obtained are used to establish the