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Suzuki, Michio Huang, Margaret Janice Fernald 2011-05-07T12:14:28Z 2011-05-07T12:14:28Z 10000-01-01 1992 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 $n.M\sb{22}.$ Made available in DSpace on 2011-05-07T12:14:28Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9305561.pdf: 7153690 bytes, checksum: b511e0831c7f74a3cbf15c4a854385dc (MD5) Previous issue date: 1992 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:34Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:16:05-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9305561 (UMI)AAI9305561 http://hdl.handle.net/2142/19660 eng Copyright 1992 Huang, Margaret Janice Fernald Mathematics Verification of the McKay-Alperin-Dade Conjecture for the covering groups of the Mathieu group M(22) text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.