Integral Group Ring ZG Research Articles

Bicyclic and Bass Cyclic Units in Group Rings

AbstractThe subgroup generated by the Bass cyclic and bicyclic units is of infinite index in the group of units of the integral group ring ZG when G is either D or

The structure of the integral grouping of a finite group with cyclic sylow subgroups

Let G be a finite group with cyclic Sylow subgroups. The integral group ring ZG is described as a multiple pullback ring, constructed from hereditary crossed product orders.

Determining Units in Some Integral Group Rings

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].

Integral group rings with trivial central units

In this note finite groups G G whose integral group ring Z G {\mathbf {Z}}G has only trivial central units are classified.

Zassenhaus conjecture forA 5

We develop a criterion for rational conjugacy of torsion units of the integral group ringℤG of a finite groupG, as also a necessary condition for an element ofℤG to be a torsion unit, and apply them to verify the Zassenhaus conjecture in case whenG=A 5.

A theorem on units of integral group rings

Let G be a finite group and U = U(ℤG) be the unit group of the integral group ring ℤG. Let H be the normal subgroup of U generated by the unit groups of the cyclic subgroup rings. It is proved that, under certain restrictions on G, H has finite index in U. The main tools are a theorem of Bass concerning K1(ℤG) and a theorem of Margulis of discrete subgroups of Lie groups.

Torsion units in integral group rings of some metabelian groups

Let G be an extension of an elementary abelian p-group A by an abelian group X with faithful and irreducible action. Then any torsion unit of the integral group ring ZG is rationally conjugate to a trivial unit. This confirms a conjecture of Zassenhaus for these groups G.

ZGn-modules, Gn cyclic of square-free order n

In the companion paper ( J. Algebra 93 (1985), 1–116) all finitely generated modules over a class of rings called Dedekind-like are described, with emphasis on the behavior of these modules in direct sums. The present paper begins by showing that the integral group ring ZG n is Dedekind-like. Some properties of ZG n -modules are studied that do not hold for Dedekind-like rings in general. The modules studied do not necessarily have torsion-free abelian groups.

On the class groups of dihedral groups

Let A be a Z-order in a semisimple Q-algebra -4. Let C(A) denote the class group of A defined by using locally free left A-modules. Define D(d) to be the kernel of the natural map C(A) + C(Q) for Q a maximal Z-order in A containing A and d(A) to be the order of D(d). The integral group ring ZG of a finite group G is a Z-order in the semisimple Q-algebra QG, and hence the groups C(ZG) and D(ZG) can be defined. The study of the groups C(ZG) and D(ZG) seems to be important, because it has applications to various problems in number theory, algebraic topology, etc. In this paper we will try, as a continuation of [3], to determine finite nonabelian groups G such that d(ZG) = 1. Denote by D, the dihedral group of order 2n and by S, , A, the symmetric, alternating group on n symbols, respectively. The following theorem has been obtained in [3].

The associated graded ring of an integral group ring

Let G be an Abelian group, the symmetric algebra of G and the associated graded ring of the integral group ring ZG, where (AG denotes the augmentation ideal of ZG). Then there is a natural epimorphism (4)which is given on the nth component byIn general θ is not an isomorphism. In fact Bachmann and Grünenfelder(1) have shown that for finite Abelian G, θ is an isomorphism if and only if G is cyclic. Thus it is of interest to investigate ker θn for finite Abelian groups. In view of proposition 3.25 of (3) it is enough to consider finite Abelian p-groups.

Integral representations of cyclic groups of order $p\sp{2}$

Let p be an odd prime, either regular or properly irregular, and let G be a cyclic group of order p2. The author determines a full set of inequivalent representations of G by matrices with rational integral entries. This information is used to calculate the ideal class number of the integral group ring ZG. Let G be a cyclic group of order p2 with generator x, where p is an odd prime. We shall determine the number n(ZG) of nonisomorphic indecompos- able ZG-lattices, for the case where p is a regular odd prime (that is, p does not divide the class number of the cyclotomic field Q (PV 1)). We shall also calculate h(ZG), the number of isomorphism classes of ideals M in ZG for which QM = QG. Such problems were considered previously by various authors (see Heller-Reiner (2), for example), but no explicit formulas for n(ZG) and h(ZG) were obtained, because of difficulties about units. The methods of Galovich (1) enable one to overcome these difficulties for the case where p is a regular odd prime. We restrict ourselves to this case from now on, unless otherwise stated. We may write the integral group ring ZG as Z (x)/(xP - 1). Letting 4pm(X) denote the cyclotomic polynomial of order p m, we now set R = Z(x)/ (p(x)), S = Z(X)/((p2(x)), T= Z(x)/(xP - 1). Then R is the ring of all algebraic integers in the cyclotomic field Q (V1 1); let

Hypercentral ideals, noetherian modules, and a theorem of Stroud

Received April 9, 1973 1. INTRODUCTION An ideal which can be generated by elements belonging to the centre of a ring R is called a central ideal of R: here and in what follows “ring” always means a ring with unity. More generally, an ideal I of R is said to be hypeycevztral if there exists an ascending central seties of ideals of R from 0 to 1, that is to say, a chain of ideals 0 = I, < II < ... I, <I&+1 < ... I, = I, such that &+i/& is a central ideal of R/I, and I,, = &,,Ia for all ordinals a < L and all limit ordinals h < 1. The least ordinal L which is the length of such a series is called the height of I. Recently Roseblade [5] proved that every ideal of the integral group ring ZG of a group G is hypercentral if and only if G is a hypercentral group, i.e., G coincides with a (possibly trans- finite) term of its upper central series. Hypercentral ideals which have finite height will be called po&central, but the reader should note that Roseblade uses this term in a narrower sense. If M is a (right unitary) R-module and S is a nonempty subset of R, let *S stand for the additive subgroup of all a in n/I such that as = 0 for all s E S. Our first object is to prove the following theorem. THEOREM 1. Let R be a ving and lM a noetheviavz R-vvwdule. Then there exists a positive integer n such that MP

Elementary Generalizations of Hilbert's Theorem 90

Let K, k be fields and K|k a finite galois extension with galois group G. The multiplicative group K* of K is a G-module, that is, a module over the integral group ring ZG, the module action of an element σ ϵ G being its effect as an automorphism. It is shown in [2, p. 158] that the first cohomology group vanishes:1