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Title:  Verification of the McKayAlperinDade Conjecture for the covering groups of the Mathieu group M(22) 
Author(s):  Huang, Margaret Janice Fernald 
Doctoral Committee Chair(s):  Suzuki, Michio 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  The McKayAlperinDade Conjecture states that the number of complex irreducible characters with a given defect d in a pblock B of a finite group G can be expressed in terms of an alternating sum of the numbers of complex irreducible characters with related defects $d\sp\prime$ in related pblocks $B\sp\prime$ of the normalizers $N\sb{G}(C)$ of representatives C of the Gconjugacy classes of radical pchains of G. Specifically, we have the following. Conjecture A (The McKayAlperinDade conjecture). If $O\sb{p}(G)$ is the Sylow psubgroup of a central subgroup N of G, and is not a defect group of B then $$\sum\limits\sb{C\in{\cal R}/G}(1)\sp{\vert C\vert}k(N\sb{G}(C),B,d,O\vert\nu) = 0$$for any $O\le Out(G\vert N),$ where $\nu$ is a linear character of N and ${\cal R}/G$ is our family of representatives. This paper presents a verification of the MAD Conjecture for the group 12.$M\sb{22},$ whose order is $2\sp9\cdot3\sp3\cdot5\cdot7\cdot11.$ Since Dade has shown that Conjecture A holds for any blocks with cyclic defect groups, this paper deals specifically with the primes 3 and 2. In each case, representatives of the $M\sb{22}$conjugacy classes of the radical psubgroups of $M\sb{22}$ are identified, together with their normalizers. Subsequently, a complete listing of the representatives of the $M\sb{22}$conjugacy classes of radical pchains C, together with their normalizers $N\sb{M\sb{22}}(C)$ is made. The normalizers $N\sb{n.M\sb{22}}(C)$ are then determined for each radical pchain C and for n = 1,2,3,4,6,12. The action of the outer automorphism group of $M\sb{22},$ which is cyclic of order 2, on each of the groups $N\sb{n.M\sb{22}}(C)$ is identified. Finally, the MAD Conjecture is verified for the 3blocks and the 2blocks of $n.M\sb{22}.$ 
Issue Date:  1992 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/19660 
Rights Information:  Copyright 1992 Huang, Margaret Janice Fernald 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9305561 
OCLC Identifier:  (UMI)AAI9305561 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics