A Remark on the Group Rings of Order Preserving Permutation Groups

1968 ◽  
Vol 11 (5) ◽  
pp. 679-680 ◽  
Author(s):  
R.H. LaGrange ◽  
A.H. Rhemtulla
Keyword(s):  
Ordered Set ◽  
Isomorphism Problem ◽  
Affirmative Answer ◽  
Permutation Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Totally Ordered Set

If G and H are two groups such that their integral group rings Z(G) and Z(H) are isomorphic, does it follow that G and H are isomorphic? This is the isomorphism problem and an affirmative answer is obtained in case G is a sub group of the group of order preserving permutations of a totally ordered set.

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 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.

scholarly journals Subgroup Isomorphism Problem for units of integral group rings

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Leo Margolis
Keyword(s):  
Finite Groups ◽  
Isomorphism Problem ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings

AbstractThe Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups

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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