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

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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Hochschild Cohomology Ring of the Mod-2 Group Ring of the Generalized Quaternion Group

Communications in Algebra ◽

10.1080/00927870600875914 ◽

2006 ◽

Vol 34 (11) ◽

pp. 3985-4005 ◽

Cited By ~ 2

Author(s):

Takao Hayami

Keyword(s):

Group Ring ◽

Hochschild Cohomology ◽

Cohomology Ring ◽

Quaternion Group ◽

Hochschild Cohomology Ring ◽

Generalized Quaternion Group ◽

Generalized Quaternion

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On Hochschild cohomology ring of the integral group ring of the quaternion group

Tsukuba Journal of Mathematics ◽

10.21099/tkbjm/1496164962 ◽

2005 ◽

Vol 29 (2) ◽

pp. 363-387 ◽

Cited By ~ 1

Author(s):

Takao Hayami

Keyword(s):

Group Ring ◽

Hochschild Cohomology ◽

Cohomology Ring ◽

Integral Group Ring ◽

Integral Group ◽

Quaternion Group ◽

Hochschild Cohomology Ring

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The Group of Units of the Integral Group Ring ZS3

Canadian Mathematical Bulletin ◽

10.4153/cmb-1972-093-1 ◽

1972 ◽

Vol 15 (4) ◽

pp. 529-534 ◽

Cited By ~ 36

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.

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

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-026-x ◽

1974 ◽

Vol 17 (1) ◽

pp. 129-130 ◽

Cited By ~ 3

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.

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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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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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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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