Unitary subgroup of the multiplicative group of the integral group ring of a cyclic group

1987 ◽  
Vol 41 (4) ◽  
pp. 265-268 ◽  
Author(s):  
A. A. Bovdi
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Unitary Subgroup

Lehmer’s problem for arbitrary groups

2020 ◽  
pp. 1-32
Author(s):  
W. Lück
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Bounded Operator ◽  
Integral Group Ring ◽  
Integral Group ◽  
Infinite Cyclic Group ◽  
Matrix Formula ◽  
Lehmer’S Problem

We consider the problem whether for a group [Formula: see text] there exists a constant [Formula: see text] such that for any [Formula: see text]-matrix [Formula: see text] over the integral group ring [Formula: see text] the Fuglede–Kadison determinant of the [Formula: see text]-equivariant bounded operator [Formula: see text] given by right multiplication with [Formula: see text] is either one or greater or equal to [Formula: see text]. If [Formula: see text] is the infinite cyclic group and we consider only [Formula: see text], this is precisely Lehmer’s problem.

Polynomial Ideals in Group Rings

1973 ◽  
Vol 25 (6) ◽  
pp. 1174-1182 ◽  
Author(s):  
M. M. Parmenter ◽  
I. B. S. Passi ◽  
S. K. Sehgal
Keyword(s):  
Commutative Ring ◽  
Free Group ◽  
Group Ring ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Polynomial Ideals

Letf(x1, x2, … , xn) be a polynomial in n non-commuting variables x1, x2, … , xn and their inverses with coefficients in the ring Z of integers, i.e. an element of the integral group ring of the free group on X1, x2, … , xn. Let R be a commutative ring with unity, G a multiplicative group and R(G) the group ring of G with coefficients in R.

scholarly journals Corresponding Group and Module Sequences

1961 ◽  
Vol 19 ◽  
pp. 27-40 ◽  
Author(s):  
R. H. Crowell
Keyword(s):  
Group Ring ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Integral Group

For convenience we consider throughout an arbitrary but fixed multiplicative group H. The integral group ring of H is denoted by ZH, and the homomorphism ε: ZH→Z is always the trivializer, or unit augmentation, defined by εh = 1 for all h ∈ H.

scholarly journals Whitehead groups of semidirect products of free groups

1978 ◽  
Vol 19 (2) ◽  
pp. 155-158 ◽  
Author(s):  
Koo-Guan Choo
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Semidirect Product ◽  
Class Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Whitehead Group ◽  
Infinite Cyclic Group ◽  
Semidirect Products ◽  
Projective Class

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].

Elementary Generalizations of Hilbert's Theorem 90

1965 ◽  
Vol 8 (6) ◽  
pp. 749-757 ◽  
Author(s):  
Ian G. Connell
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Cohomology Group ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
First Cohomology Group ◽  
Hilbert’S Theorem 90 ◽  
Image Position

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

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