Revisiting the Zassenhaus conjecture on torsion units for the integral group rings of small groups

2015 ◽  
Vol 125 (2) ◽  
pp. 167-172 ◽  
Author(s):  
ALLEN HERMAN ◽  
GURMAIL SINGH
Keyword(s):  
Small Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Zassenhaus Conjecture ◽  
Torsion Units

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 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.

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

scholarly journals On a conjecture of Zassenhaus on torsion units in integral group rings. II

1986 ◽  
Vol 97 (2) ◽  
pp. 201-201 ◽  
Author(s):  
C{ésar Polcino Milies ◽  
J{ürgen Ritter ◽  
Sudarshan K. Sehgal
Keyword(s):  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Torsion Units

scholarly journals Torsion units in integral group rings of some metabelian groups, II

1987 ◽  
Vol 25 (3) ◽  
pp. 340-352 ◽  
Author(s):  
Z Marciniak ◽  
J Ritter ◽  
S.K Sehgal ◽  
A Weiss
Keyword(s):  
Group Rings ◽  
Integral Group ◽  
Metabelian Groups ◽  
Integral Group Rings ◽  
Torsion Units
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