The associated graded ring of an integral group ring

1977 ◽  
Vol 82 (1) ◽  
pp. 25-33 ◽  
Author(s):  
Inder Bir S. Passi ◽  
Lekh Raj Vermani
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Symmetric Algebra ◽  
Integral Group Ring ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Finite Abelian Groups ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Image Position

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.

On the Structure of Q2(G) for Finitely Generated Groups

1973 ◽  
Vol 25 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Lower Central Series ◽  
Integral Group Ring ◽  
Symmetric Power ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Finitely Generated ◽  
Finitely Generated Groups

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

On polynomial groups

1970 ◽  
Vol 68 (2) ◽  
pp. 285-289 ◽  
Author(s):  
L. R. Vermani
Keyword(s):  
Group Ring ◽  
Abelian Groups ◽  
Integral Group Ring ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Image Position

If M is a group, Z(M) its integral group ring and AM the augmentation ideal, then following Passi we can form the Abelian groups

The projective extensions of a finite group

1981 ◽  
Vol 90 (2) ◽  
pp. 251-257
Author(s):  
P. J. Webb
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Projective Module ◽  
Integral Group Ring ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Definition Of ◽  
Image Position ◽  
A Finite Group ◽  
Exact Sequences

Let G be a finite group and let g be the augmentation ideal of the integral group ring G. Following Gruenberg(5) we let (g̱) denote the category whose objects are short exact sequences of zG-modules of the form and in which the morphisms are commutative diagramsIn this paper we describe the projective objects in this category. These are the objects which satisfy the usual categorical definition of projectivity, but they may also be characterized as the short exact sequencesin which P is a projective module.

Polynomial functors

1969 ◽  
Vol 66 (3) ◽  
pp. 505-512 ◽  
Author(s):  
I. B. S. Passi
Keyword(s):  
Group Ring ◽  
Abelian Groups ◽  
Integral Group Ring ◽  
Recent Theory ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Polynomial Functors ◽  
Derived Functors ◽  
Image Position

1. Introduction: If G is a group, Z(G) its integral group-ring and AG the augmentation ideal, then we can form the Abelian groupsIn (5) we have studied the structure of these Abelian groups which we called polynomial grouups. If C denotes the category of Abelian groups, then Pn and Qn are functors from C into C. We call these functors polynomial functors. The object of this work is to study the nature of these funtors. Except for n = 1, these functors are non-additive. In fact, in the sense of Eilenberg–Maclane (4) these are functors of degree exactly n (Theorem 2·3). Because of their non-additive nature, their derived functors cannot be calculated in the traditional Cartan–Eilenberg(1) method. We have to make use of the more recent theory of Dold–Puppe (3).

Natural maps of extension functors and a theorem of R. G. Swan

1961 ◽  
Vol 57 (3) ◽  
pp. 489-502 ◽  
Author(s):  
P. J. Hilton ◽  
D. Rees
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Group Ring ◽  
Projective Module ◽  
Additive Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Natural Maps ◽  
Image Position ◽  
A Finite Group

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.

On the Structure of Augmentation Quotient Groups for the Generalized Quaternion Group

2012 ◽  
Vol 19 (01) ◽  
pp. 137-148 ◽  
Author(s):  
Qingxia Zhou ◽  
Hong You
Keyword(s):  
Group Ring ◽  
Integral Group Ring ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Quaternion Group ◽  
Generalized Quaternion Group ◽  
Augmentation Quotient ◽  
Generalized Quaternion ◽  
Quotient Groups

For the generalized quaternion group G, this article deals with the problem of presenting the nth power Δn(G) of the augmentation ideal Δ (G) of the integral group ring ZG. The structure of Qn(G)=Δn(G)/Δn+1(G) is obtained.

The Group of Units of the Integral Group Ring ZS3

1972 ◽  
Vol 15 (4) ◽  
pp. 529-534 ◽  
Author(s):  
I. Hughes ◽  
K. R. Pearson
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Torsion Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
List Type ◽  
Type Number ◽  
Quaternion Group ◽  
Group Of Units

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is(i) abelian and the order of each element divides 4, or(ii) abelian and the order of each element divides 6, or(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

A TRANSFINITE FILTRATION OF SCHUR MULTIPLICATOR

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1061-1073
Author(s):  
ROMAN MIKHAILOV ◽  
INDER BIR S. PASSI
Keyword(s):  
Group Ring ◽  
Lower Central Series ◽  
Integral Group Ring ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Relationship Of ◽  
The Relationship

We study certain subgroups of the Schur multiplicator of a group G. These subgroups are related to the identification of subgroups of G determind by ideals in its integral group ring ℤ[G]. Suitably defined transfinite powers of the augmentation ideal of ℤ[G] provide an increasing transfinite filtration of the Schur multiplicator of G. We investigate the relationship of this filtration with the transfinite lower central series of groups which are HZ-local in the sense of Bousfield.

N-Series and Filtrations of the Augmentation Ideal

1974 ◽  
Vol 26 (4) ◽  
pp. 962-977 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Augmentation Ideal ◽  
Free Basis ◽  
Image Position

Let G be a group. Denote by ZG the group ring of G over the integers and by Δ = Δ(G) the augmentation ideal of ZG, that is, the kernel of the augmentation map ϵ : ZG → Z defined by . Then Δ is a free abelian group with a free basis . A filtration of Δ is a sequence

Généralisation d'un Lemme de Kummer

1989 ◽  
Vol 32 (4) ◽  
pp. 486-489 ◽  
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Finite Abelian Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Ring Endomorphism ◽  
Mod P

AbstractIf A is a finite abelian group and ZA its integral group ring, consider units u ∊ ZA which have coefficient sum = 1 and are fixed under the involution a —> a-1, a ∊ A. For an odd regular prime p and a p-group A, it is shown that u ≡ 1 mod p if only if u = π(v)v-p, where v is the same kind of unit, and π is the ring endomorphism given by a —> ap, a∊A.

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