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Title:Verification of Dade's conjecture for Janko group J(,3)
Author(s):Kotlica, Sonja
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Dade has 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.
Dade claims that the most complicated form of this conjecture, called the "Inductive Conjecture", will hold for all finite groups if it holds for all covering groups of finite simple groups. In this thesis I verify the inductive conjecture for all covering groups of the third Janko group $J\sb3$ (in the notation of the Atlas of Finite Groups). This is one step in the inductive proof of the conjecture for all finite groups.
The Schur Multiplier of $J\sb3$ is cyclic of order 3. Hence there are just two covering groups of $J\sb3,$ namely $J\sb3$ itself and a central extension $3.J\sb3$ of $J\sb3$ by a cyclic group Z of order 3. I treat these two covering groups separately.
The outer automorphism group Out($J\sb3)$ of $J\sb3$ is cyclic of order 2. In this case Dade affirms in that the Inductive Conjecture for $J\sb3$ is equivalent to the much weaker "Invariant Conjecture." Furthermore, Dade has proved that this Invariant Conjecture holds for all blocks with cyclic defect groups. The Sylow p-subgroups of $J\sb3$ are cyclic of order p for all primes dividing $\vert J\sb3\vert$ except 2 and 3. So I verify the Invariant Conjecture for the two primes $p=2$ and $p=3.$
The group Out(3.$J\sb3{\mid}Z)$ of outer automorphisms of 3.$J\sb3$ centralizing Z is trivial. In this case Dade affirms that the Inductive Conjecture is equivalent to the "Projective Conjecture." Again, Dade has shown that this Projective Conjecture holds for all blocks with cyclic defect groups. So I verify the Projective Conjecture for $p=2$ and $p=3.$
Issue Date:1996
Rights Information:Copyright 1996 Kotlica, Sonja
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9712336
OCLC Identifier:(UMI)AAI9712336

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