The Augmentation Terminals of Certain Locally Finite Groups

Let G be a group and ZG be the integral group ring of G. We shall write 𝔤 for the augmentation ideal of G; that is to say, the kernel of the homomorphism of ZG onto Z which sends each group element to 1. The powers gλ of 𝔤 are defined inductively for ordinals λ by 𝔤λ = 𝔤μ𝔤, if λ = μ + 1, and otherwise. The first ordinal λ for which gλ = 𝔤λ+1 is called the augmentation terminal or simply the terminal of G. For example, if G is either a cyclic group of prime order or else isomorphic with the additive group of rational numbers then gn > 𝔤ω = 0 for all finite n, so that these groups have terminal ω.The groups with finite terminal are well-known and easily described. If G is one such, then every homomorphic image of G must also have finite terminal.

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Milnor-Witt K-Theory and Tensor Powers of the Augmentation Ideal of ℤ[F×]

Algebra Colloquium ◽

10.1142/s1005386713000485 ◽

2013 ◽

Vol 20 (03) ◽

pp. 515-522 ◽

Cited By ~ 1

Author(s):

Kevin Hutchinson ◽

Liqun Tao

Keyword(s):

Group Ring ◽

Additive Group ◽

Tensor Algebra ◽

Augmentation Ideal ◽

Ideal Formula ◽

K Theory ◽

Tensor Powers

Let F be a field of characteristic not equal to 2. We describe the relation between the non-negative dimensional Milnor-Witt K-theory of F and the tensor algebra over the group ring ℤ[F×] of the augmentation ideal [Formula: see text]. In the process, we clarify the structure of the additive group [Formula: see text], giving a simple presentation in particular.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100035544 ◽

1961 ◽

Vol 57 (3) ◽

pp. 489-502 ◽

Cited By ~ 24

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.

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

Algebra Colloquium ◽

10.1142/s1005386712000090 ◽

2012 ◽

Vol 19 (01) ◽

pp. 137-148 ◽

Cited By ~ 1

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.

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BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

International Journal of Algebra and Computation ◽

10.1142/s0218196711006327 ◽

2011 ◽

Vol 21 (04) ◽

pp. 531-545 ◽

Cited By ~ 3

Author(s):

JAIRO Z. GONÇALVES ◽

ÁNGEL DEL RÍO

Keyword(s):

Finite Group ◽

Free Group ◽

Group Ring ◽

Prime Order ◽

Infinite Order ◽

Integral Group Ring ◽

Integral Group ◽

Cyclic Unit ◽

Central Elements ◽

Group A

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1973-034-4 ◽

1973 ◽

Vol 25 (2) ◽

pp. 353-359 ◽

Cited By ~ 17

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.

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A Class of infinite soluble groups with an A-group condition

Glasgow Mathematical Journal ◽

10.1017/s001708950000402x ◽

1980 ◽

Vol 21 (1) ◽

pp. 81-84

Author(s):

M. J. Tomkinson

Keyword(s):

Homomorphic Image ◽

Finite Groups ◽

Locally Finite ◽

Soluble Groups ◽

Sylow Subgroups ◽

Locally Finite Groups ◽

Group Condition

Finite soluble groups in which all the Sylow subgroups are abelian were first investigated by Taunt [8] who referred to them as A-groups. Locally finite groups with the same property have been considered by Graddon [2]. By the use of Sylow theorems it is clear that every section (homomorphic image of a subgroup) of an A-group is also an A-group and hence every nilpotent section of an A-group is abelian. This is the characterization that we use here in considering groups which are not, in general, periodic.

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The associated graded ring of an integral group ring

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100053640 ◽

1977 ◽

Vol 82 (1) ◽

pp. 25-33 ◽

Cited By ~ 5

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.

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On a bottom layer in a group

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS ◽

10.31489/2020m4/136-142 ◽

2020 ◽

Vol 100 (4) ◽

pp. 136-142

Author(s):

V.I. Senashov ◽

◽

I.A. Paraschuk ◽

◽

Keyword(s):

Finite Elements ◽

Finite Number ◽

Finite Index ◽

Finite Groups ◽

Prime Order ◽

Bottom Layer ◽

Arbitrary Group ◽

Locally Finite

We consider the problem of recognizing a group by its bottom layer. This problem is solved in the class of layer-finite groups. A group is layer-finite if it has a finite number of elements of every order. This concept was first introduced by S. N. Chernikov. It appeared in connection with the study of infinite locally finite p-groups in the case when the center of the group has a finite index. S. N. Chernikov described the structure of an arbitrary group in which there are only finite elements of each order and introduced the concept of layer-finite groups in 1948. Bottom layer of the group G is a set of its elements of prime order. If have information about the bottom layer of a group we can receive results about its recognizability by bottom layer. The paper presents the examples of groups that are recognizable, almost recognizable and unrecognizable by its bottom layer under additional conditions.

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A TRANSFINITE FILTRATION OF SCHUR MULTIPLICATOR

International Journal of Algebra and Computation ◽

10.1142/s0218196705002785 ◽

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.

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