Bicyclic and Bass cyclic units in group rings

Manuscripta Mathematica, 1992

For any finite abelian group A, let U(A) denote the multiplicative group of the integral group ring ZA. In his 1965 paper [1], Bass showed how to construct generators for a free abelian subgroup of finite index in U(A). More recently, another such construction was described in [7], with the advantage of producing subgroups of much lower index in U(A), but the disadvantage of being constrained to p-groups A of odd order. The aim of this paper is to extend the second procedure to arbitrary A, and to clarify its relation to the first one. The Bass units are obtained by a kind of surrogate "division" within ZA of (x-1)" by (y-1) m, where z 6 A and y 6 A each generate the same cyclic group C C_ A, and m > 1 is an appropriate integer. The procedure of [7] depends on first constructing a suitable integral polynomial w(X), and then using the set of units {w(z) I z 6 A} to generate a subgroup of U(A). For odd p-groups A, this sometimes yields "all" units (modulo the trivial units :hA), and quite often "almost all" (especially for regular p). By contrast, the finite index of the Bass construction tends to be huge. For instance, if IAI = 67, this index is greater than 1056 , while the other method still gives all units. In the present paper, we start with a cyclic group C. Putting G = Aut(C), and fixing a generator z of C, we construct a canonical G-map w : A2(G)~ U(C), where A2(G) is the square of the kernel A(G) of the obvious ring surjection (coefficient sum): ZG-, Z. In fact, if H denotes G modulo its subgroup of order 2, our map induces an isomorphism w: ,~(H) ~, W(C) C U(C), whose image W(C) is our basic building block. The group B,,(C) generated by the Bass units attached to C happens to be the w-image of the submodule mZl(H) c_ zl2(g). The unit w(x) which plays the lead role in [7] is simply the image of a generator for the principal ideal A2(H) in the p-group case. As C ranges over all cyclic subgroups of a given A, the product of the W(C)-which is direct-forms a free subgroup of finite index in U(A).

Indian Journal of Pure and Applied Mathematics, 2021

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group U (ZG) of the integral group ring ZG of a finite group G. These constructions rely on explicit constructions of units in ZG and proofs of main results make use of the description of the Wedderburn components of the rational group algebra QG. The latter relies on explicit constructions of primitive central idempotents and the rational representations of G. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.

2009

Extending an idea of Bass, one can construct a large torsion-free group Y(A) of units in the integral group ring ZZA of any finite abelian group A. This group of constructible units is “good” in two ways: it easily allows the selection of explicit sets of independent generators (bases), and it is typically of low (often trivial) index in the the full unit group of ZZA. The present paper deals with cases where Y(A) does not suffice to describe the full unit group, but where it is nevertheless possible to produce bases of the latter by using constructible units together with co-induced ones. AMS Subject Classifications: 11T22, 20C05. This work was supported by an operating grant from NSERC (Canada) and by the hospitality of SFB 343 at the University of Bielefeld (Germany).

Mathematics of Computation, 2013

We give a constructive proof of the theorem of Bass and Milnor saying that if G is a finite abelian group then the Bass units of the integral group ring ZG generate a subgroup of finite index in its unit group U(ZG). Our proof provides algorithms to represent some units that contribute to only one simple component of QG and generate a subgroup of finite index in U(ZG) as product of Bass units. We also obtain a basis B formed by Bass units of a free abelian subgroup of finite index in U(ZG) and give, for an arbitrary Bass unit b, an algorithm to express b ϕ(|G|) as a product of a trivial unit and powers of at most two units in this basis B.

Communications in Algebra, 1999

In this note we give a description of the central units of an integral group ring 'llG for an arbitrary group G. We also give a set of generators of a subgroup of finite index in the centre of the unit group when G is any group whose FC-centre is finitely generated. Jespers, Parmenter and Sehgal did the same for finitely generated nilpotent groups.

Manuscripta Mathematica, 1993

arXiv (Cornell University), 2020

The augmentation powers in an integral group ring ZG induce a natural filtration of the unit group of ZG analogous to the filtration of the group G given by its dimension series {D n (G)} n≥1. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.

Journal of Group Theory, 2007

For an arbitrary group G, and a G-adapted ring R (for example, R ¼ Z), let U be the group of units of the group ring RG, and let Z y ðUÞ denote the union of the terms of the upper central series of U, the elements of which are called hypercentral units. It is shown that Z y ðUÞ c N U ðGÞ. As a consequence, hypercentral units commute with all unipotent elements, and if G has non-normal finite subgroups with RðGÞ denoting their intersection, then ½U; Z y ðUÞ c RðGÞ. Further consequences are given as well as concrete examples.

Journal of Group Theory, 2005

We introduce generic units in ZC n and prove that they are precisely the shifted cyclotomic polynomials. They generate the group YðC n Þ of constructible units. For each cyclic group we produce a basis of a finite index subgroup of integral units consisting of certain irreducible cyclotomic polynomials; this extends a result of Hoechsmann and Ritter. We also study 'alternating-like' units and decide when they generate a subgroup of finite index.

2020

The aim of this article is to draw attention towards various natural but unanswered questions related to the lower central series of the unit group of an integral group ring.

This thesis deals with the problem of describing the unit group of specific group rings over the integers. Some results are given in an expository manner as the proofs of the results are normally very dependent on the particular group. The ones presented by this method later on are S3,D4,D6 and A4. Next, we present the method for groups of order p3, where p is an odd prime. These come from a paper by Ritter and Sehgal. I consider both non-abelian groups of order p3 and descriptions of the unit groups of both of their respective group rings are presented.I present the method as applied to the groups of order 27. The last theoretical results are on determining the unit group of the group ring over a group of order pq where p ≡ 1( mod q). These results come from a paper by Luthar [3]. No practical examples are done of this method. The next part deals with presenting actual groups and determining the unit structure of their integral group ring. The first two, S3 and D4 are from previous authors. The first was done by Hughes and Pearson [2], the second by C. Polcino- Milies [4]. I also present, D6, which is new result of the thesis.

Journal of Algebra, 2014

Let G be a finite group, u a Bass unit based on an element a of G of prime order, and assume that u has infinite order modulo the center of the units of the integral group ring ZG. It was recently proved that if G is solvable then there is a Bass unit or a bicyclic unit v and a positive integer n such that the group generated by u n and v n is a non-abelian free group. It has been conjectured that this holds for arbitrary groups G. To prove this conjecture it is enough to do it under the assumption that G is simple and a is a dihedral p-critical element in G. We first classify the simple groups with a dihedral p-critical element. They are all of the form PSL(2, q). We prove the conjecture for p = 5; for p > 5 and q even; and for p > 5 and q + 1 = 2p. We also provide a sufficient condition for the conjecture to hold for p > 5 and q odd. With the help of computers we have verified the sufficient condition for all q < 10000.

Boletim da Sociedade Brasileira de Matemática, 1973

There are very few cases known of nonabelian groups G where the group of central units of ZG, denoted Z(U (ZG)), is nontrivial and where the structure of Z(U (ZG)), including a complete set of generators, has been determined. In this note, we show that the central units of augmentation 1 in the integral group ring ZA 5 form an infinite cyclic group u , and we explicitly find the generator u.

Groups, rings, and group rings, 2006

There are very few cases known of nonabelian groups G where the group of central units of ZG, denoted Z(U (ZG)), is nontrivial and where the structure of Z(U (ZG)), including a complete set of generators, has been determined. In this note, we show that the central units of augmentation 1 in the integral group ring ZA 5 form an infinite cyclic group u , and we explicitly find the generator u.

Journal of Number Theory, 1984

It is proved that if G is a split extension of a cyclic-p-group by a cyclic p'-group with faithful action then any torsion unit of augmentation one of LG is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (JA 1,1X1) = 1 then any torsion unit of 9G of augmentation one and order relatively prime to IA / is rationally conjugate to an element of X.