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.

On the Torsion Units of Some Integral Group Rings

2006 ◽  
Vol 13 (02) ◽  
pp. 329-348 ◽  
Author(s):  
Martin Hertweck
Keyword(s):  
Finite Group ◽  
Finite Simple Group ◽  
Fundamental Result ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Torsion Unit ◽  
Zassenhaus Conjecture ◽  
A Finite Group ◽  
Torsion Units

It is shown that any torsion unit of the integral group ring ℤG of a finite group G is rationally conjugate to a trivial unit if G = P ⋊ A with P a normal Sylow p-subgroup of G and A an abelian p′-group (thus confirming a conjecture of Zassenhaus for this particular class of groups). The proof is an application of a fundamental result of Weiss. It is also shown that the Zassenhaus conjecture holds for PSL(2,7), the finite simple group of order 168.

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.

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 Torsion units in integral group rings of metacyclic groups

1984 ◽  
Vol 19 (1) ◽  
pp. 103-114 ◽  
Author(s):  
César Polcino Milies ◽  
Sudarshan K. Sehgal
Keyword(s):  
Group Rings ◽  
Integral Group ◽  
Metacyclic Groups ◽  
Integral Group Rings ◽  
Torsion Units
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