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].

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 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.

scholarly journals On the prime graph question for almost simple groups with an alternating socle

2017 ◽  
Vol 27 (03) ◽  
pp. 333-347 ◽  
Author(s):  
Andreas Bächle ◽  
Mauricio Caicedo
Keyword(s):  
Simple Group ◽  
Group Ring ◽  
Prime Graph ◽  
Integral Group Ring ◽  
Simple Groups ◽  
Alternating Group ◽  
Integral Group ◽  
Almost Simple Group ◽  
Almost Simple Groups

Let [Formula: see text] be an almost simple group with socle [Formula: see text], the alternating group of degree [Formula: see text]. We prove that there is a unit of order [Formula: see text] in the integral group ring of [Formula: see text] if and only if there is an element of that order in [Formula: see text] provided [Formula: see text] and [Formula: see text] are primes greater than [Formula: see text]. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle [Formula: see text] if [Formula: see text].

scholarly journals On the prime graph question for integral group rings of 4-primary groups I

2017 ◽  
Vol 27 (06) ◽  
pp. 731-767 ◽  
Author(s):  
Andreas Bächle ◽  
Leo Margolis
Keyword(s):  
Prime Graph ◽  
Affirmative Answer ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Almost Simple Groups ◽  
Integral Group Rings ◽  
Primary Groups ◽  
Composition Factors

We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to [Formula: see text] for [Formula: see text], establishing the Prime Graph Question for all groups where the only non-abelian composition factors are of the aforementioned form. Using this, we determine exactly how far the so-called HeLP method can take us for (almost simple) groups having an order divisible by at most four different primes.

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

2021 ◽  
Vol 28 (04) ◽  
pp. 561-568
Author(s):  
Jinke Hai ◽  
Lele Zhao
Keyword(s):  
Abelian Group ◽  
Simple Group ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
Coleman Automorphism

Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

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.

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.

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.

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