Integral group ring of the first Janko simple group (original) (raw)

Torsion units in integral group rings of Janko simple groups

Mathematics of Computation, 2010

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.

Integral group ring of the Suzuki sporadic simple group

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.

Kimmerle conjecture for the Held and O'Nan sporadic simple groups

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.

Integral group ring of the Mathieu simple group M 12

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.

Groups with the same orders of Sylow normalizers as the Janko groups

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,

Broué’s Abelian defect group conjecture holds for the Janko simple group

Journal of Pure and Applied Algebra, 2008

In the representation theory of finite groups, there is a well known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an Abelian defect group P, then A and its Brauer corresponding block B of the normalizer N G (P) of P in G are equivalent (Rickard equivalent). This conjecture is called Broué's Abelian defect group conjecture. We prove in this paper that Broué's Abelian defect group conjecture is true for a non-principal 3-block A with an elementary Abelian defect group P of order 9 of the Janko simple group J 4 . It then turns out that Broué's Abelian defect group conjecture holds for all primes p and for all p-blocks of the Janko simple group J 4 .

Broué’s Abelian defect group conjecture holds for the Janko simple group J 4

Journal of Pure and Applied Algebra, 2008

In the representation theory of finite groups, there is a well known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an Abelian defect group P, then A and its Brauer corresponding block B of the normalizer N G (P) of P in G are equivalent (Rickard equivalent). This conjecture is called Broué's Abelian defect group conjecture. We prove in this paper that Broué's Abelian defect group conjecture is true for a non-principal 3-block A with an elementary Abelian defect group P of order 9 of the Janko simple group J 4 . It then turns out that Broué's Abelian defect group conjecture holds for all primes p and for all p-blocks of the Janko simple group J 4 .

Verification of Dade's Conjecture for Janko GroupJ3

Journal of Algebra, 1997

In 3 Dade made a conjecture expressing the number k B, d of characters of a given defect d in a given p-block B of a finite group G in terms of the Ž. corresponding numbers k b, d for blocks b of certain p-local subgroups of G. w x Several different forms of this conjecture are given in 5. Dade claims that the most complicated form of this conjecture, called the w x ''Inductive Conjecture 5.8'' in 5 , will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive Ž conjecture for all covering groups of the third Janko group J in the notation of 3 w x. the Atlas 1. This is one step in the inductive proof of the conjecture for all finite groups. Certain properties of J simplify our task. The Schur Multiplier of J is cyclic of 3 3 Ž w x. order 3 see 1, p. 82. Hence, there are just two covering groups of J , namely J 3 3 itself and a central extension 3 и J of J by a cyclic group Z of order 3. We treat 3 3 these two covering groups separately. Ž. Ž w x. The outer automorphism group Out J of J is cyclic of order 2 see 1, p. 82. 3 3 w x In this case Dade affirms in 5, Section 6 that the inductive conjecture for J is 3 w x equivalent to the much weaker ''Invariant Conjecture 2.5'' in 5. Furthermore, w x Dade has proved in 6 that this Invariant Conjecture holds for all blocks with cyclic defect groups. The Sylow p-subgroups of J are cyclic of order p for all primes 3 < < dividing J except 2 and 3. So we only need to verify the Invariant Conjecture for 3 the two primes p s 2 and p s 3. We do that in Theorem 2.10.1 and Theorem 3.6.1 below. Ž <. The group Out 3 и J Z of outer automorphisms of 3 и J centralizing Z is trivial 3 3 Ž w x. see 1, p. 82. In this case Dade affirms that the Inductive Conjecture is w x w x equivalent to the ''Projective Conjecture 4.4'' in 5. Again, Dade has shown in 6 that this projective conjecture holds for all blocks with cyclic defect groups. So we only need to verify the projective conjecture for p s 2 and p s 3. We do that in Theorem 4.4.2 and Theorem 4.3.1.