Verification of Dade's Conjecture for Janko GroupJ3

Abstract

In [3] Dade made a conjecture expressing the numberk(B,d) of characters of a given defectdin a givenp-blockBof a finite groupGin terms of the corresponding numbersk(b,d) for blocksbof certainp-local subgroups ofG. Several different forms of this conjecture are given in [5].Dade claims that the most complicated form of this conjecture, called the “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 conjecture for all covering groups of the third Janko groupJ3(in the notation of the Atlas [1]). This is one step in the inductive proof of the conjecture for all finite groups.Certain properties ofJ3simplify our task. The Schur Multiplier ofJ3is cyclic of order 3 (see [1, p. 82]). Hence, there are just two covering groups ofJ3, namelyJ3itself and a central extension 3·J3ofJ3by a cyclic groupZof order 3. We treat these two covering groups separately.The outer automorphism group Out(J3) ofJ3is cyclic of order 2 (see [1, p. 82]). In this case Dade affirms in [5, Section 6] that the inductive conjecture forJ3is equivalent to the much weaker “Invariant Conjecture 2.5” in [5]. Furthermore, Dade has proved in [6] that this Invariant Conjecture holds for all blocks with cyclic defect groups. The Sylowp-subgroups ofJ3are cyclic of orderpfor all primes dividing |J3| except 2 and 3. So we only need to verify the Invariant Conjecture for the two primesp=2 andp=3. We do that in Theorem 2.10.1 and Theorem 3.6.1 below.The group Out(3·J3|Z) of outer automorphisms of 3·J3centralizingZis trivial (see [1, p. 82]). In this case Dade affirms that the Inductive Conjecture is 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 forp=2 andp=3. We do that in Theorem 4.4.2 and Theorem 4.3.1.For the above reasons the four Theorems 2.10.1, 3.6.1, 4.4.2, and 4.3.1 below are sufficient to prove the inductive form of Dade's conjecture for all covering groups ofJ3.This paper is my Ph.D. thesis. It would not have been possible without the help and infinite patience of my advisor Everett Dade.