Free Groups and Subgroups of Finite Index in the Unit Group of an Integral Group Ring

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-Abelian group G for which every nonlinear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary Abelian 2-group.     

Integral Group Ring of the Mathieu Simple Group M 23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M ₂₃ using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M ₁₁ and M ₁₂. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.     

交错群A_5与二面体群D_6直积的整群环的挠单位

本文利用Luthar-Passi方法,研究了五次交错群A_5与六阶二面体群D_6直积的整群环的挠单位,得到了该群的Zassenhaus猜想成立.     

On the Structure of the Units of Group Algebra of Dihedral Group

In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.     

Rank Two Integral Representations of the Infinite Dihedral Group

We give an effective classification of the representations of the infinite dihedral group in GL ₂(R) where R is either the valuation ring ?_(p) or the ring of p-adic integers.     

Torsion Units in the Integral Group Ring of the Alternating Group of Degree 6

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring ?G of a finite group G conjugates to a group element within the rational group algebra ?G. We investigate the Zassenhaus Conjecture (ZC) and a conjecture by W. Kimmerle about prime graph in the normalized unit group of ?A₆.     

Unit Groups of Integral Finite Group Rings with No Noncyclic Abelian Finite p-Subgroups

It is shown that in the units of augmentation one of an integral group ring ? G of a finite group G, a noncyclic subgroup of order p ², for some odd prime p, exists only if such a subgroup exists in G. The corresponding statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed by Kimmerle.     

The Normalizer Property for Integral Group Rings of a Class of Complete Monomial Groups

Let G = H?S _m be the natural wreath product of H by S _m , where H is a finite 2-closed group and S _m is the symmetric group of degree m. It is shown that the normalizer property holds for G.     

On the Commuting Graph of Dihedral Group

Let Γ be a non-abelian group and Ω ? Γ. We define the commuting graph G = 𝒞(Γ, Ω) with vertex set Ω and two distinct elements of Ω are joined by an edge when they commute in Γ. In this article, among some properties of commuting graphs, we investigate distant properties as well as detour distant properties of commuting graph on D_2n. We also study the metric dimension of commuting graph on D_2n and compute its resolving polynomial.     

The Subgroup Normalizer Problem for Integral Group Rings of Some Nilpotent and Metacyclic Groups

For a group G and a subgroup H of G, this article discusses the normalizer of H in the units of a group ring RG. We prove that H is only normalized by the “obvious” units, namely products of elements of G normalizing H and units of RG centralizing H, provided H is cyclic. Moreover, we show that the normalizers of all subgroups of certain nilpotent and metacyclic groups in the corresponding group rings are as small as possible. These classes contain all dihedral groups, all finite nilpotent groups, and all finite groups with all Sylow subgroups being cyclic.     

Hochschild Cohomology Ring of the Mod-2 Group Ring of the Generalized Quaternion Group

We will determine the ring structure of the Hochschild cohomology HH?( ₂ Q _t ) of the mod-2 group ring  ₂ Q _t for arbitrary generalized quaternion groups Q _t of order 4t by calculating the ordinary cup product in H?(Q _t , _ψ  ₂ Q _t ).     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Multiplicative Jordan Decomposition in Integral Group Ring of Group K_8×C_5

In this article,we present the multiplicative Jordan decomposition in integral group ring of group K8 × C5,where K8 is the quaternion group of order 8.Thus,we give a positive answer to the question raised by Hales A W,Passi I B S and Wilson L E in the paper "The multiplicative Jordan decomposition in group rings II.     

主理想环上子群Gr在线性群中的扩群

Suppose R is a principal ideal ring,R~* is a multiplicative group which is composed of all reversible elements in R,and M_n(R),GL(n,R),SL(n,R) are denoted by, M_n(R)={A=(a_(ij))_(n×n)|a_(ij)∈R,i,j=1,2,…,n},GL(n,R) = {g|g∈M_n(R),detg∈R~*},SL(n,R) = {g∈GL(n,R)|det g=1},SL(n,R)≤G≤GL(n,R)(n≥3),respectively, then basing on these facts,this paper mainly focus on discussing all extended groups of G_r={(AB OD)∈G|A∈GL(r,R),(1≤r相似文献   

On Certain Quotients of Lower Central Factors of a Normal Subgroup of a Free Group

We compute subgroups of the normal subgroup R of a free group F determined by certain ideals contained in the augmentation ideal Δ(R) and then prove certain subquotients of R to be free Abelian.     

单位上三角矩阵群的注记

记Tr_1(n,Z)是整数环Z上对角线元素全是1的全体上三角矩阵组成的群,k_(ij)(1≤i相似文献   

Some Results on Hypercentral Units in Integral Group Rings

Abstract

In this note we investigate the hypercentral units in integral group rings ?G,where G is not necessarily torsion. One of the main results obtained is the following (Theorem 3.5): if the set of torsion elements of G is a subgroup T of G and if Z ₂(𝒰) is not contained in C _𝒰(T),then T is either an Abelian group of exponent 4 or a Q* group. This extends our earlier result on torsion group rings.     

一类域上的线性变换环的群结构

In this paper, we introduce a practical method for obtaining the structure of the group of units for the ring of linear transformations of a vector space over an arbitrary field, and we give a further generalization of the result in [3].