scholarly journals Free products in the unit group of the integral group ring of a finite group

2017 ◽  
Vol 145 (7) ◽  
pp. 2771-2783
Author(s):  
Geoffrey Janssens ◽  
Eric Jespers ◽  
Doryan Temmerman
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Unit Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Free Products ◽  
A Finite Group

scholarly journals Units of group rings of groups of order 16

1993 ◽  
Vol 35 (3) ◽  
pp. 367-379 ◽  
Author(s):  
E. Jespers ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Abelian Groups ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Group Of Units ◽  
Abelian Case ◽  
A Finite Group

LetGbe a finite group,(ZG) the group of units of the integral group ring ZGand1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively(ZG) for particular groupsG.This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described(ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of(ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

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.

Embedding free products in the unit group of an integral group ring

2004 ◽  
Vol 82 (2) ◽  
pp. 97-102 ◽  
Author(s):  
J. Z. Gon�alves ◽  
D. S. Passman
Keyword(s):  
Group Ring ◽  
Unit Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Free Products

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.

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.

On the vanishing of twisted nil groups

2008 ◽  
Vol 3 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Daniel Juan-Pineda ◽  
Rafael Ramos
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
A Finite Group ◽  
Free Order

AbstractLet G be a finite group and [G] its integral group ring. We prove that the twisted nil groups N([G]) vanish for all i ≤ 1 for G a finite group of square-free order.

ON THE UNITS OF THE INTEGRAL GROUP RING Z[G × Cp]

2008 ◽  
Vol 07 (03) ◽  
pp. 393-403 ◽  
Author(s):  
RICHARD M. LOW
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Direct Product ◽  
Group Ring ◽  
Open Problem ◽  
Structure Theorem ◽  
Integral Group Ring ◽  
Integral Group ◽  
Group Of Units ◽  
A Finite Group

Describing the group of units U(ZG) of the integral group ring ZG, for a finite group G, is a classical and open problem. In this paper, it is shown that U(Z[G × Cp]) = M ⋊ U(ZG), a semi-direct product where M is a certain subgroup of U(Z[ζ]G) and p prime. For p = 2, this structure theorem is applied to give precise descriptions of U(ZG) for a non-abelian group G of order 32, G = C10, and G = C8 × C2.

scholarly journals INTEGRAL GROUP RING OF THE MATHIEU SIMPLE GROUP M24

2012 ◽  
Vol 11 (01) ◽  
pp. 1250016 ◽  
Author(s):  
VICTOR BOVDI ◽  
ALEXANDER KONOVALOV
Keyword(s):  
Simple Group ◽  
Group Ring ◽  
Unit Group ◽  
Positive Answer ◽  
Integral Group Ring ◽  
Integral Group ◽  
Sporadic Group ◽  
Prime Graphs ◽  
Zassenhaus Conjecture

We study the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M24. As a consequence, for this group we give a positive answer to the question by Kimmerle about prime graphs.

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