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Verification of the McKay-Alperin-Dade Conjecture for the covering groups of the Mathieu group M(22)

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Title: Verification of the McKay-Alperin-Dade 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 Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
Abstract: The McKay-Alperin-Dade Conjecture states that the number of complex irreducible characters with a given defect d in a p-block 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 p-blocks $B\sp\prime$ of the normalizers $N\sb{G}(C)$ of representatives C of the G-conjugacy classes of radical p-chains of G. Specifically, we have the following.Conjecture A (The McKay-Alperin-Dade conjecture). If $O\sb{p}(G)$ is the Sylow p-subgroup 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 M-A-D 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 p-subgroups 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 p-chains 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 p-chain 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 M-A-D Conjecture is verified for the 3-blocks and the 2-blocks 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: 2011-05-07
Identifier in Online Catalog: AAI9305561
OCLC Identifier: (UMI)AAI9305561
 

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