Abstract
The dimension subgroup problem asks whether &(Z, G) = G, , the n-th term of the lower central series of G. Results on this problem have been set down in [ll, 121. This paper investigates the subgroups for the general ring R. I f the dimension subgroup conjecture is true, this investigation is successful; that is, it is possible to express I,(R, G) in terms of certain canonical subgroups of G and certain dimension subgroups in(Zpe , G), where Z,. is the ring of integers modulo the prime powerp”. The expression depends upon the arithmetic of the ring R. The analysis of &,(R, G), carried out in Section 2, is largely a study of this arithmetic. The first section examines the modular dimension subgroups i, (Zve, G). Using Lie-theoretic methods, Lazard calculated these subgroups for free groups. His subgroups are characterized here group-theoretically; they are not generally the appropriate subgroups but are seen to be closely related. Next a systematic exposition is given of an old method of finding sets of generators for the powers of the augmentation ideal. Results of Jennings and of Lazard on Z,G are established. The section concludes with the calculation of the first three modular dimension subgroups. Included here is a new proof of the dimension subgroup conjecture for class 2 groups. The method used could solve simultaneously both the dimension subgroup problem and the integral group ring problem
Published Version
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