Benson's cofibrants, Gorenstein projectives and a related conjecture

2021 ◽  
pp. 1-21
Author(s):  
Rudradip Biswas
Keyword(s):  
Group Ring ◽  
Commutative Rings ◽  
Global Dimension ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Short Article ◽  
Finite Global Dimension

Abstract In this short article, we will be principally investigating two classes of modules over any given group ring – the class of Gorenstein projectives and the class of Benson's cofibrants. We begin by studying various properties of these two classes and studying some of these properties comparatively against each other. There is a conjecture made by Fotini Dembegioti and Olympia Talelli that these two classes should coincide over the integral group ring for any group. We make this conjecture over group rings over commutative rings of finite global dimension and prove it for some classes of groups while also proving other related results involving the two classes of modules mentioned.

On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

1990 ◽  
Vol 42 (3) ◽  
pp. 383-394 ◽  
Author(s):  
Frank Röhl
Keyword(s):  
Group Ring ◽  
Modular Group ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Affirmative Answer ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

Trivial Units in Group Rings

2000 ◽  
Vol 43 (1) ◽  
pp. 60-62 ◽  
Author(s):  
Daniel R. Farkas ◽  
Peter A. Linnell
Keyword(s):  
Recent Result ◽  
Group Ring ◽  
Finite Index ◽  
Integral Group Ring ◽  
Alternative Proof ◽  
Crystallographic Group ◽  
Group Rings ◽  
Integral Group ◽  
Arbitrary Group ◽  
Normalized Units

AbstractLet G be an arbitrary group and let U be a subgroup of the normalized units in ℤG. We show that if U contains G as a subgroup of finite index, then U = G. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS

2011 ◽  
Vol 10 (04) ◽  
pp. 711-725 ◽  
Author(s):  
J. Z. GONÇALVES ◽  
D. S. PASSMAN
Keyword(s):  
Group Ring ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Nonabelian Group ◽  
Ring Of Integers ◽  
Integral Group Rings ◽  
Noncyclic Subgroup

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk, m(x), uk, m(x*)) or (uk, m(x), uk, m(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

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.

Polynomial Ideals in Group Rings

1973 ◽  
Vol 25 (6) ◽  
pp. 1174-1182 ◽  
Author(s):  
M. M. Parmenter ◽  
I. B. S. Passi ◽  
S. K. Sehgal
Keyword(s):  
Commutative Ring ◽  
Free Group ◽  
Group Ring ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Polynomial Ideals

Letf(x1, x2, … , xn) be a polynomial in n non-commuting variables x1, x2, … , xn and their inverses with coefficients in the ring Z of integers, i.e. an element of the integral group ring of the free group on X1, x2, … , xn. Let R be a commutative ring with unity, G a multiplicative group and R(G) the group ring of G with coefficients in R.

scholarly journals Integral group rings of solvable groups with trivial central units

2018 ◽  
Vol 30 (4) ◽  
pp. 845-855 ◽  
Author(s):  
Andreas Bächle
Keyword(s):  
Solvable Group ◽  
Group Ring ◽  
Solvable Groups ◽  
Integral Group Ring ◽  
Finite Solvable Group ◽  
Group Rings ◽  
Integral Group ◽  
Frobenius Groups ◽  
Integral Group Rings ◽  
Rational Groups

Abstract The integral group ring {\mathbb{Z}G} of a group G has only trivial central units if the only central units of {\mathbb{Z}G} are {\pm z} for z in the center of G. We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.

scholarly journals Units and augmentation powers in integral group rings

2020 ◽  
Vol 23 (6) ◽  
pp. 931-944
Author(s):  
Sugandha Maheshwary ◽  
Inder Bir S. Passi
Keyword(s):  
Present Article ◽  
Group Ring ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Natural Filtration

AbstractThe augmentation powers in an integral group ring {\mathbb{Z}G} induce a natural filtration of the unit group of {\mathbb{Z}G} analogous to the filtration of the group G given by its dimension series {\{D_{n}(G)\}_{n\geq 1}}. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.

Trivial Units for Group Rings with G-adapted Coefficient Rings

2005 ◽  
Vol 48 (1) ◽  
pp. 80-89 ◽  
Author(s):  
Allen Herman ◽  
Yuanlin Li ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Finite Characteristic

AbstractFor each finite group G for which the integral group ring ℤG has only trivial units, we give ring-theoretic conditions for a commutative ring R under which the group ring RG has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if R is a ring of finite characteristic and RG has only trivial units, then G has order at most 3.

Units in Integral Group Rings for Order pq

1995 ◽  
Vol 47 (1) ◽  
pp. 113-131
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Finite Index ◽  
Finite Abelian Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Cyclotomic Units

AbstractFor any finite abelian group A, let Ω(A) denote the group of units in the integral group ring which are mapped to cyclotomic units by every character of A. It always contains a subgroup Y(A), of finite index, for which a basis can be systematically exhibited. For A of order pq, where p and q are odd primes, we derive estimates for the index [Ω(A) : Y(A)]. In particular, we obtain conditions for its triviality.

scholarly journals On the Gruenberg–Kegel graph of integral group rings of finite groups

2017 ◽  
Vol 27 (06) ◽  
pp. 619-631 ◽  
Author(s):  
W. Kimmerle ◽  
A. Konovalov
Keyword(s):  
Group Ring ◽  
Finite Groups ◽  
Prime Graph ◽  
Unit Group ◽  
Integral Group Ring ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Almost Simple Groups ◽  
Integral Group Rings

The prime graph question asks whether the Gruenberg–Kegel graph of an integral group ring [Formula: see text], i.e. the prime graph of the normalized unit group of [Formula: see text], coincides with that one of the group [Formula: see text]. In this note, we prove for finite groups [Formula: see text] a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups [Formula: see text] whose order is divisible by at most three primes and show that the Gruenberg–Kegel graph of such groups coincides with the prime graph of [Formula: see text].

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