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

2013 ◽  
Vol 20 (03) ◽  
pp. 515-522 ◽  
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.

The Augmentation Terminals of Certain Locally Finite Groups

1972 ◽  
Vol 24 (2) ◽  
pp. 221-238 ◽  
Author(s):  
K. W. Gruenberg ◽  
J. E. Roseblade
Keyword(s):  
Homomorphic Image ◽  
Group Ring ◽  
Finite Groups ◽  
Prime Order ◽  
Additive Group ◽  
Rational Numbers ◽  
Group Element ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Locally Finite

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.

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.

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.

ℵ0-Injective group rings

2019 ◽  
Vol 19 (03) ◽  
pp. 2050042
Author(s):  
Liang Shen
Keyword(s):  
Group Ring ◽  
Natural Generalization ◽  
Group Rings ◽  
Ideal Formula ◽  
Strongly Connected

Recall that a ring [Formula: see text] is called right [Formula: see text]-injective if every homomorphism from a countably generated right ideal of [Formula: see text] to [Formula: see text] can be extended to a homomorphism from [Formula: see text] to [Formula: see text]. These rings are not only a natural generalization of self-injective rings but also strongly connected with regularities of rings. Let [Formula: see text] be the group ring of a group [Formula: see text] over a ring [Formula: see text]. It is proved that [Formula: see text] is right [Formula: see text]-injective if and only if (i) [Formula: see text] is right [Formula: see text]-injective; (ii) [Formula: see text] is finite; (iii) for each countably generated right ideal [Formula: see text] of [Formula: see text], any right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text] can be extended to a right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text].

Lie Powers and Pseudo-Idempotents

2011 ◽  
Vol 54 (2) ◽  
pp. 297-301 ◽  
Author(s):  
Marianne Johnson ◽  
Ralph Stöhr
Keyword(s):  
Symmetric Group ◽  
Direct Summand ◽  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
Ground Field ◽  
Tensor Powers

AbstractWe give a new factorisation of the classical Dynkin operator, an element of the integral group ring of the symmetric group that facilitates projections of tensor powers onto Lie powers. As an application we show that the iterated Lie power L2(Ln) is a module direct summand of the Lie power L2n whenever the characteristic of the ground field does not divide n. An explicit projection of the latter onto the former is exhibited in this case.

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