Verification of Dade's conjecture for Janko group J(3)

[摘要] 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 corresponding numbers k(b, d) for blocks b of certain p-local subgroups of G. 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 group J(3) (in the notation of the Atlas [1]). This is one step in the inductive proof of the conjecture for all finite groups. Certain properties of J(3) simplify our task. The Schur Multiplier of J(3) is cyclic of order 3 (see [1, p. 82]). Hence, there are just two covering groups of J(3), namely J(3) itself and a central extension 3 . J(3) Of J(3) by a cyclic group Z of order 3. We treat these two covering groups separately. The outer automorphism group Out(J(3)) of J(3) is cyclic of order 2 (see [1, p. 82]). In this case Dade affirms in [5, Section 6] that the inductive conjecture for J(3) is 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 Sylow p-subgroups of J(3) are cyclic of order p for all primes dividing \J(3)\ except 2 and 3. So we only need to verify the Invariant Conjecture for the two primes p = 2 and p = 3. We do that in Theorem 2.10.1 and Theorem 3.6.1 below. The group Out(3 . J(3)\Z) of outer automorphisms of 3 . J(3) centralizing Z is 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 for p = 2 and p = 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 of J(3). This paper is my Ph.D. thesis. It would not have been possible without the help and infinite patience of my advisor Everett Dade. (C) 1997 Academic Press.

[发布日期] 1997-01-15