scholarly journals On the prime graph question for integral group rings of Conway simple groups

2019 ◽  
Vol 95 ◽  
pp. 162-176
Author(s):  
Leo Margolis
Keyword(s):  
Prime Graph ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings

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].

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.

scholarly journals On the Structure of Integral Group Rings of Sporadic Groups

2000 ◽  
Vol 3 ◽  
pp. 274-306 ◽  
Author(s):  
Frauke M. Bleher ◽  
Wolfgang Kimmerle
Keyword(s):  
Automorphism Groups ◽  
Symmetric Groups ◽  
Computational Procedure ◽  
Isomorphism Problem ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Sporadic Groups ◽  
Integral Group Rings ◽  
Sporadic Simple Groups

AbstractThe object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

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.

scholarly journals Bass units as free factors in integral group rings of simple groups

2014 ◽  
Vol 404 ◽  
pp. 100-123 ◽  
Author(s):  
Jairo Z. Gonçalves ◽  
Robert M. Guralnick ◽  
Ángel del Río
Keyword(s):  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings

scholarly journals Torsion units in integral group rings of Janko simple groups

2010 ◽  
Vol 80 (273) ◽  
pp. 593-615 ◽  
Author(s):  
V. A. Bovdi ◽  
E. Jespers ◽  
A. B. Konovalov
Keyword(s):  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Torsion Units

scholarly journals TORSION UNITS IN INTEGRAL GROUP RINGS OF CONWAY SIMPLE GROUPS

2011 ◽  
Vol 21 (04) ◽  
pp. 615-634 ◽  
Author(s):  
V. A. BOVDI ◽  
A. B. KONOVALOV ◽  
S. LINTON
Keyword(s):  
Unit Group ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Torsion Units

Using the Luthar–Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups Co 1, Co 2 and Co 3.

scholarly journals Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

2017 ◽  
Vol 60 (4) ◽  
pp. 813-830 ◽  
Author(s):  
Andreas Bächle ◽  
Leo Margolis
Keyword(s):  
Prime Graph ◽  
Affirmative Answer ◽  
New Method ◽  
Group Rings ◽  
Integral Group ◽  
Modular Representation Theory ◽  
Integral Group Rings ◽  
Zassenhaus Conjecture ◽  
Normalized Units ◽  
Torsion Units

AbstractWe introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus conjecture for the group PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

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