University of Illinois Urbana-Champaign

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

Huang, Margaret Janice Fernald

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Description

Title
Verification of the McKay-Alperin-Dade Conjecture for the covering groups of the Mathieu group M(22)
Author(s)
Huang, Margaret Janice Fernald
Issue Date
1992
Doctoral Committee Chair(s)
Suzuki, Michio
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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 $n.M\sb{22}.$
Type of Resource
text
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http://hdl.handle.net/2142/19660
Copyright and License Information
Copyright 1992 Huang, Margaret Janice Fernald

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