Torsion units in integral group rings of Janko simple groups

arXiv (Cornell University), 2006

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups J 1 , J 2 and J 3 is the same as that of the normalized unit group of their respective integral group ring.

2008

Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerle's conjecture on prime graphs.

Proceedings of the American Mathematical Society, 1986

Suppose that a group G has a normal subgroup C where C and G/C are cyclic of relatively prime orders. Then any torsion unit in ZG is rationally conjugate to a trivial unit.

Using the Luthar-Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture. Date: November 3rd, 2008. 1991 Mathematics Subject Classification. Primary 16S34, 20C05, secondary 20D08.

Rendiconti del Circolo Matematico di Palermo, 2007

We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M 12 . As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.

Indian Journal of Pure and Applied Mathematics, 2021

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group U (ZG) of the integral group ring ZG of a finite group G. These constructions rely on explicit constructions of units in ZG and proofs of main results make use of the description of the Wedderburn components of the rational group algebra QG. The latter relies on explicit constructions of primitive central idempotents and the rational representations of G. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.

International Journal of Algebra and Computation, 2011

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 Co1, Co2and Co3.

1996

This is a short survey in which some questions related to the Zassenhaus Conjecture on finite subgroups in integral group rings are discussed. The bibliography is incomplete but gives a possibility to surch for more references and to find out the details of the subject development .

J. Appl. Algebra Discrete Struct, 2005

There exist many characterizations for the sporadic simple groups. In this paper we give two new characterizations for the Mathieu sporadic groups. Let M be a Mathieu group and let p be the greatest prime divisor of |M|. In this paper, we prove that M is uniquely determined by |M| and |N M (P)|, where P ∈ Syl p (M). Also we prove that if G is a finite group, then G ∼ = M if and only if for every prime q,

2002

We present a survey of some recent results on problems posed by Sudarshan Sehgal. In this paper some new results are presented on the following problems posed by Sudarshan Sehgal in [43] . The integral group ring of a group G is denoted by ZG. Its unit group we denote by U(ZG), its group of normalized units by UI (ZG) and its center by Z(U(ZG» . Problem 17: Give a presentation by generators and relations for the unit group U(ZG) of the integral group ring ZG for some finite groups G . Problem 23: Give generators up t.o finite index for U(ZG) if G is a finite group. Problem 18: Find a good estimate for the index (U1 (7l.. G) : B) , where B is the group generated by the Bass cyclic units and the bicyclic units. Problem 19: Is the group generated by the bicylic units torsion-free? Problem 29: Suppose that G is a finite nilpotent group . Does G have a normal complement N in UI (Z G) , i.e., UI (ZG) = N >4 G? Problem 43: Let G be a finite group. Is Nu(zG)(G) = G Z(U(ZG))? In the late ...