Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups

Abstract

This paper solves for torsion free and torsion abelian groups G the problem of presenting nth powers D n ðGÞ of the augmentation ideal DðGÞ of an integral group ring ZG; in terms of the standard additive generators of D n ðGÞ: A concrete basis for D n ðGÞ is obtained when G itself has a basis and is torsion. The results are applied to describe the homology of the sequence D n ðNÞG:D n ðGÞ7D n ðG=NÞ: r

The Augmentation Quotients of the Groups of Order 25, II

“…So, we may assume that i ≥ n + 7, and that u − 1 i ∈ S n for all i satisfying n ≤ i < i. By (2) In the case h = 1, we prove (d) by induction on i. If n − 1 ≤ i ≤ n + 2, it is clear that P ∈ S n .…”

“…Then G has a uniqueness basis X = x y y 4 with respect to the integer sequence (4,4,2) where x = y = 1 and y 4 = 2.…”

“…Then = 10, and s n = 11 for all n ≥ 74. THE GROUP G 22In this section, letG = G 22 = u z u 16 = z 2 = 1 uz = zu 9Then X = z u u 8 is a uniqueness basis for G 22 with respect to the integer sequence(2,8,2) where z = u = 1 and u 8 = 2. Downloaded by [Michigan State University] at 00:48 28 December 2014…”

Structure of augmentation quotients of finite homocyclic abelian groups

“…However it is usually difficult to write down explicitly a basis of Δ n (G) for an arbitrary finite abelian group, even for the abelian p-group cases. It is noted that in [8] Bak and the author gave a procedure for constructing such an explicit basis. However, that kind of bases is too complicated to be used in our purpose now in practice.…”

On the structure of the augmentation quotient group for some nonabelian 2-groups

“…much work has been done [1], [2], [4], [5], [6], [7]. In [4], Hales and Passi (see also [1]) proved that for a finite abelian group G there exists a number N such that for all n N , Q n (G) is isomorphic to Q N (G).…”