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Uni International 300 N INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1.The sign or “target” for pages apparently lacking from the document photographed is “Missing Page(s)”. If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method of “sectioning” the material has been followed. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. Uni International 300 N. Zeeb Road Ann Arbor, Ml 48106 8426500 Woldar, Andrew J. ON THE MAXIMAL SUBGROUPS OF LYONS’ GROUP AND EVIDENCE FOR THE EXISTENCE OF A 111-DIMENSIONAL FAITHFUL LYS-MODULE OVER A FIELD OF CHARACTERISTIC 5 The Ohio State University Ph.D. 1984 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 ON THE MAXIMAL SUBGROUPS OF LYONS' GROUP AND EVIDENCE FOR THE EXISTENCE OF A 111-DIMENSIONAL FAITHFUL LyS-MODULE OVER A FIELD OF CHARACTERISTIC 5 DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Andrew J. Woldar, B.S., M.S. The Ohio State University 1984 Reading Committee: Approved by Ronald Solomon Koichiro Harada Sia K. Wong Department of Mathematics Dedicated to My Mother and the Memory of My Father George Morris Woldar (1910-1964) ACKNOWLEDGEMENTS I wish to express my gratitude to Richard Lyons and Peter Landrock for their helpful and insightful sug­ gestions, and particularly to my adviser, Ronald Solomon, for his guidance throughout the work. VITA December 12, 1947 Born - New York, New York 1976 B.S., The City College of New York, C.U.N.Y., New York, New York 1976-1984 Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio 1978. M.S., The Ohio State University, Columbus, Ohio 1976,1982,1983 Recipient of Summer Fellow­ ship, The Ohio State University, Columbus, Ohio 1983 Recipient of Graduate Student Alumni Research Award FIELDS OF STUDY Major Field: Mathematics Studies in Group Theory. Professor Ronald Solomon Studies in Representation Theory. Professor Ronald Solomon TABLE OF CONTENTS Page ACKNOWLEDGEMENTS....................................iii VITA ................................................. iv LIST OF TABLES ...................................... vi LIST OF FIGURES...................................... vii LIST OF SYMBOLS..................................... viii INTRODUCTION .......................................... 1 CHAPTER I. THE MAXIMAL LOCAL SUBGROUPS OF LYONS’ GROUP. 6 1. 2-Local Analysis ........................ 18 2. 3-Local Analysis ............ 35 3. 5-Local Analysis ........................ 52 4. Local Analysis for Remaining Primes. 84 5. A Complete Set of Maximal Local Subgroups for LyS. .................. 8 6 II. ON THE MAXIMAL SUBGROUPS OF LYONS’ GROUP . 89 1. Some Non-Local Analysis....................91 2. Conclusions............... 127 III. EVIDENCE FOR A 111-DIMENSIONAL IRREDUCIBLE LyS-MODULE OVER A FIELD OF CHARACTERISTIC 5 . 146 1. 5-Modular Analysis of M e ................ 152 2. The 5-Decomposition Matrices for 2n (5 ^ n ^ 11) and A x x ..................... 172 3. The Character Tables and 5-Decomposition Matrices for N = E 3S *Mii and S = 3 2 + lt-(SL(2,5)*Z8) ..................... 210 4. Evidence .................................. 237 5. Minimal Degree for a Faithful Character of L y S .....................................263 LIST OF REFERENCES ................................. 270 v LIST OF TABLES Table Page 1. Centralizers of Prime Order Elements .... 14 2. The Character Induction-Restriction Table for (Me, 1^ ( 3 ) ) .......................... 154 3. The Character Induction-Restriction Table for (Mc,M22) ............................ 1 5 5 4. The Character Induction-Restriction Table for (.3,Me) .............................. 1 6 9 5. 5-Cores of Partitions of Size n ( 5 ^ n ^ l l ) ..................................... 1 7 4 6 . The Character Table for N = E 35»M1 1 ........... 214 7. The Character Table for S =3 2+lf • (SL(2 , 5)*Za ). 218 LIST OF FIGURES Figure Page 1. 2-Local Structure of Ly S .............. 15 2. 3-Local Structure of LyS ........ 16 3. 5-Local Structure of Ly S .............. 17 vii LIST OF SYMBOLS A is a subgroup of B the elements of A not contained in B split extension of A by B non-split extension of A by B direct product of A and B central product of A and B the subgroup generated by the elements Xj,f ..Xjj of some group K the group generated by the subgroups A lf..,An of some group K the element y-1xy of K where x , y K the element x_1xy of K where x , y K the subgroup <[x,y]> of K where x ranges over A and y ranges over B (We call [A,A] the commutator subgroup or , derived group of A.) the index of B in A the normalizer of H in K the centralizer of H in K the centralizer of x in K (When K = LyS , we write N_,(H) , Cr (H) , and Cr(x) for N„(H) , C„(H) , and C„(x) respectively.) viii Z(K) the center of K Op(K) the largest normal p-subgroup of K 0(K) the largest normal odd-order subgroup of K ^(K) the subgroup of the p-group K generated by its elements of order p ®(K) the Prattini subgroup of K K* the non-identity elements of K |K| the order of K (We call x a p-element if | <x> | = pn for some prime p and positive integer n.) |K|p the order of a Sylow p-subgroup of K ir(K) the set of prime divisors of | K | (We call K a p -group if p is a prime not contained in tt(K). Also we call x a p -ele­ ment - or p-regular - if p is not contained in ir(<x>). ) x ~ K y x is K-conjugate to y (When K = LyS we omit the subscript "K" .) XK the set of all K-conjugates of x SP Sylow p-subgroup Frob(n,m) Frobenius group with kernel of order n and complement of order m (q = pn) Elementary abelian p-group of order q symmetric group of degree n alternating group of degree n the two-fold cover of A X1 (in the sense of Schur) 2 { i ^ , . , ijj} symmetric group on the letters {ij,..,in} A{i^ f ••>in) alternating group on the letters { i j i n} Ln (q) PSL(n,q), the group of projective special n x n matrices with entries in GF(q) Un (q) PSU(n,q), the group of projective special n x n unitary matrices with entries in GF(q2) SU±(2,5) group of 2 x 2 unitary matrices with entries in GF(25) and having determinant ±1 Me the sporadic simple group of McLaughlin of order 2 7 »36 *53 *7*11 Me the triple cover of Me (in the sense of Schur) Janko's 2nd sporadic simple group, of order 2 7.3 3 .5 2 . 7 ^11 ’^12 » 2 the three smallest of the five sporadic simple groups of Mathieu GF(q) (q = pn ). the Galois field of q elements «a+b special p-group of order p a+b with center of order pa (When a = l we call pa+k extraspecial.) (X»4»)k the inner product (defined by the standard orthogonality relations) applied to the characters x anc* f of K X+H the restriction of x to H where x is a character of an over-group of H X+M the character x induced to M where x is a character of a subgroup of M % the principal character of K Irr(K) the set of ordinary irreducible characters of K IBrp (K) the set of irreducible p-Brauer charac­ ters of K ik (0) the inertia group in K of the character 0 X < * X is a constituent of ij; deg(x) the degree of the character x xi INTRODUCTION With little doubt, one of the most significant ad­ vances in modern algebra is the classification of finite simple groups. Believed to be at a state of completion, the classification represents the culmination of a century- long endeavor, riddled with great periods of inactivity and skepticism. We elaborate briefly on it, so as to establish a context for the work which follows. The classification reveals the existence of seventeen infinite families of finite simple groups: sixteen families of "finite simple groups of Lie type" and the alternating groups.
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