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 Remark on the Units of Finite Order in The Group Ring of a Finite Group

1974 ◽  
Vol 17 (1) ◽  
pp. 129-130 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Finite Group ◽  
Finite Order ◽  
Direct Product ◽  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
Quaternion Group ◽  
Group Of Units ◽  
A Finite Group

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will proveLet G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

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.

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.

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.

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 second augmentation quotient of an integral group ring

1978 ◽  
Vol 31 (1) ◽  
pp. 259-265 ◽  
Author(s):  
A. W. Hales ◽  
I. B. S. Passi
Keyword(s):  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
Augmentation Quotient
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