Units of group rings of groups of order 16

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).

Canadian Mathematical Bulletin, 1990

In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].

2017

One of the main problems on group rings is to determine its group of units. In this paper, we describe the group of units of integral group rings of two extra-special 2-groups: one of order 32, the central product of two copies of D4, and another of order 128, the central product of three copies of D4. 2010 Mathematics Subject Classifications: 16S34, 16U60, 20D15

Journal of Pure and Applied Algebra, 1996

In the first part we give a survey of some recent results on constructing finitely many generators for a subgroup of finite index in the unit group of an integral group ring

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.

Proceedings of the American Mathematical Society, 2011

In this paper we give new constructions of central units that generate a subgroup of finite index in the central units of the integral group ring Z G \mathbb {Z} G of a finite group. This is done for a very large class of finite groups G G , including the abelian-by-supersolvable groups.

Communications in Algebra, 2005

2021

For a finite group G and U: = U(ℤG), the group of units of the integral group ring of G, we study the implications of the structure of G on the abelianization U/U' of U. We pose questions on the connections between the exponent of G/G' and the exponent of U/U' as well as between the ranks of the torsion-free parts of Z(U), the center of U, and U/U'. We show that the units originating from known generic constructions of units in ℤG are well-behaved under the projection from U to U/U' and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ^+18].

Manuscripta Mathematica, 1995

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.

Transactions of the American Mathematical Society, 2005

We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.

2019

The fixed point properties and abelianization of arithmetic subgroups Γ of SLn(D) and its elementary subgroup En(D) are well understood except in the degenerate case of lower rank, i.e. n = 2 and Γ = SL2(O) with O an order in a division algebra D with only a finite number of units. In this setting we determine property (FA) for E2(O) and its subgroups of finite index. Along we construct a generic and computable exact sequence describing E2(O). We also consider En(R) for n ≥ 3 and R an arbitrary finitely generated ring for which we give a short algebraic proof of the existence of a global fixed point when acting on (n−2)-dimensional CAT(0) cell complexes. The latter is called property (FAn−2). Thenceforth, we investigate applications in integral representation theory. We obtain a group and ring theoretical characterization of when the unit group U(ZG) of the integral group ring of a finite group G satisfies property (T). A crucial step for this is a reduction to arithmetic groups SLn...

Advances in Mathematics, 2007

We classify the finite groups G such that the group of units of the integral group ring ZG has a subgroup of finite index which is a direct product of free-by-free groups.

2020

Let G be a finite group and U (Z G) the unit group of the integral group ring Z G. We prove a unit theorem, namely a characterization of when U(ZG) satisfies Kazhdan's property (T), both in terms of the finite group G and in terms of the simple components of the semisimple algebra QG. Furthermore, it is shown that for U( Z G) this property is equivalent to the weaker property FAb (i.e. every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in U (Z G) have both finite abelianization and are not a non-trivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SL_n(O), where O is an order in a finite dimensional semisimple Q-algebra D, and finite groups G which have the so-called cut property. For such groups G we describe the simple epimorphic images of Q G. The proof of the unit theorem fundame...

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.