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 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.

Augmentation quotients of some nonabelian finite groups

1979 ◽  
Vol 85 (2) ◽  
pp. 261-270 ◽  
Author(s):  
Gerald Losey ◽  
Nora Losey
Keyword(s):  
Group Structure ◽  
Lower Central Series ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Nilpotent Residual ◽  
Periodic Behaviour ◽  
Augmentation Quotient ◽  
Image Position ◽  
A Finite Group

1. LetGbe a group,ZGits integral group ring and Δ = ΔGthe augmentation idealZGBy anaugmentation quotientofGwe mean any one of theZG-moduleswheren, r≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotientsQn(G) =Qn,1(G) and(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determineQn(G) andPn(G) for finiteGit is sufficient to assume thatGis ap-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelianp-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: ifGis a finite group then there exist natural numbersn0and π such thatQn(G) ≅Qn+π(G) for alln≥n0; ifGωis the nilpotent residual ofGandG/Gωhas classcthen π divides l.c.m. {1, 2, …,c}. There do not appear to be any examples in the literature of this periodic behaviour forc> 1. One of goals here is to present such examples. These examples will be from the class of finitep-groups in which the lower central series is anNp-series.

scholarly journals The class of a nilpotent wreath product

1977 ◽  
Vol 17 (1) ◽  
pp. 53-89 ◽  
Author(s):  
David Shield
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Group Ring ◽  
Upper Bound ◽  
Lower Central Series ◽  
Nilpotency Class ◽  
Central Series ◽  
Augmentation Ideal ◽  
Cambridge Philos ◽  
Base Group

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

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.

Subgroups dual to dimension subgroups

1972 ◽  
Vol 71 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Robert Sandling
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Modular Group ◽  
Lower Central Series ◽  
Wreath Products ◽  
Central Series ◽  
Group Rings ◽  
Augmentation Ideal ◽  
Integral Group ◽  
New Criterion

Associated with, the powers of the augmentation ideal are the dimension subgroups. In the integral group ring case, they have long been conjectured to be the terms of the lower central series. This paper investigates the subgroups associated with the chain of ideals dual to the chain of powers of the augmentation ideal. The study is reduced to the case of the modular group rings of p-groups. The subgroups are calculated for Abelian p-groups, p odd. They appear in the upper central series of wreath products and provide a new criterion for the nilpotence of an arbitrary wreath product. The nilpotence class of wreath products is considered here as well; calculations and bounds are given; in particular, a new method of computing the class of the Sylow p-subgroups of the symmetric group arises.

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 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.

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