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Microfilms International 300 N /EE B ROAD INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help you understand 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 you of complete continuity. 2. When an image on the film is obliterated with a round black mark it is an indication that the film inspector noticed either blurred copy because of movement during exposure, or duplicate copy. 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ANN ARBOR, Ml 48106 10 BEDEORD ROW. LONDON WC 1 R 4EJ, ENGLAND 8001817 Ro t h , Ro b er t Lyle, j r . HALL TRIPLE SYSTEMS AND COMMUTATIVE MOUFANG EXPONENT 3 LOOPS The Ohio State University PH.D. 1979 University Microfilms International 300 N. Zeeb Road, Aim Aitooi, MI 48106 18 Bedford Row, London WC1R 4EJ. England HALL TRIPLE SYSTEMS AND COMMUTATIVE MOUFANG EXPONENT 3 LOOPS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Robert Lyle Roth, Jr., B.S., M.Sc, ***** The Ohio State University 1979 Reading Committee; Approved By Professor D. K. Ray-Chaudhuri Professor E. Bannai Professor T. A. Dowling ^ i ~* Professor G. N. Robertson Professor J. C. D. S. Yaqub Adviser Department of Mathematics This work is dedicated to my mother and to the memory of my father. Their love will be my treasure always. il ACKNOWLEDGEMENTS The subject of this dissertation was suggested by my adviser, Professor D. K. Ray-Chaudhuri. I wish to express my gratitude for his helpful suggestions, his motivating questions, and his encouragement in the preparation of this work, in addition I want to thank him for the many excellent courses and seminars he has presented - they have given me the chance to see and learn some of the beauty of combinatorial mathematics. Part of Professor Ray-Chaudhuri’s time spent on discussion and reading related to this dissertation was supported by NSF grant number MCS75-08231-A01. I also want to thank Professors J. C. D. S. Yaqub, A. P. Sprague, and J. Ferrar for their kindness in discussing various aspects of this work with me. I am grateful to professors T. A. Dowling, R. M. Wilson, E. Bannai, G. N* Robertson, and R. p. Gupta for the training they have provided. I also want to acknowledge these excellent teachers who have shared a part of their knowledge with m e : Professors J. P. Tull, R. Gold, and A. Ross at The Ohio State University and Professors N. C. Ankeny, J. Munkres, G. B. Thomas, A. P. Mattuck, E. Speer, and G. Sacks at the Massachusetts Institute of Technology. ill My appreciation goes to Professors D. Kelly and M. Splvak who sparked my first interest in becoming a mathematician. I am indebted to Mrs. D. Shapiro and Mrs. M. Greene for their excellent technical assistance in the preparation of this treatise. I want to express my heartfelt thanks to my wife-to-be Kathy for her kindness, patience, and loving support. iv VITA May 8, 1952........ Born - Jacksonville, Florida 197^ ............. B.S., Massachusetts Institute of Technology, Cambridge, Massachusetts 1 9 7 ^ - 1 9 7 5 . University Fellow, Department of Mathematics, The Ohio State University, Columbus, Ohio 1975-1977.......... Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio 1976 • * ......... M.Sc., The Ohio State University, Columbus, Ohio 1977-197 8......... University Fellow, The Ohio State University, Columbus, Ohio 1978-197 9.......... Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio FIELD OF STUDY Major Field: Mathematics Studies in Combinatorial Theory. Professor D. K. Ray-Chaudhuri v TABLE OF CONTENTS Page DEDICATION.............................................. II ACKNOWLEDGEMENTS .................................. iii VITA .................................................... v LIST OF T A B L E S ............................................ viii LIST OF F I G U R E S ........................................ ix INTRODUCTION .................................. ..... 1 Chapter I. PRELIMINARIES; HISTORY AND DEFINITIONS......... 6 Conventions and Basics Hall's Original Work Characterization of Affine Spaces Perfect Matroid Designs Coordinatization by Loops HTS’s in New Contexts II. AN ACCOUNT OF BRUCK'S THEORY OF MOUFANG LOOPS . 79 Moufang Loops Commutative Moufang Loops New Contributions to the Theory III. THE BRUCK CONSTRUCTION......................... ldA The Loops Bd and The Malbos Variation IV. COMMUTATIVE MOUFANG EXPONENT 3 LOOPS OF NILFOTENCE CLASS 2 .................................. 135 The Behavior of the Product The Free Construction The Construction of Loops with Specified Properties vi Page V. THE CATALOGUE OF HALL TRIPLE SYSTEMS OF CARDINALITY AT MOST ^ , . ................... I78 • 1 li General Remarks and the Cases |G| - 3 and IGI-35 The Case |g | = 3 VI. COMMUTATIVE MOUFANG EXPONENT 3 LOOPS OF NILPOTENCE CLASS 3 ............................... 203 The Behavior of the Product The Construction of F . VII. MISCELLANEOUS RESULTS............................. 236 Designs Derived from HTS's Automorphism Groups Concluding Remarks BIBLIOGRAPHY................... 261+ vii LIST OF TABIES Page Table 1. R(m1,m2) ...................................... Ill 2. Dependencies among the Generating Basic 2-Associators in a c-SPf exp-3 Loop of Dimension ^ and Nilpotence Class 3 ....... ......... 189 3 . Equivalence of 6- t u p l e s ......................... 195 U. Equivalence of 4-tuples ....................... 201 5 . The Blocks of Dfi(H8l) ........................... 251 viii LIST OF FIGURES Page Figure 1. Pasch A x i o m ....................................... 25 2. The Loop Based at e ............................. 'jh 3* The Loop of a 3 - N e t ............................... 58 Reflection of 1-Lines........................ 59 5 . Reflection and the C o r e ........................... 90 ix INTRODUCTION The purpose of this dissertation is to study a class of * Steiner Triple Systems called Hall Triple Systems (abbreviated HTS*s) which are very special with regard to a geometric property they possess. The sets of axioms for affine spaces that have been used include a "three-dimensional" axiom. Fur example, in the axiomatization due to Lenz [37] it is the transitivity of parallelism and in the axiomatization used by Sasaki in [53] it is the require­ ment that a non-empty intersection of two distinct planes contained in a 3-flat must be a line. It is natural to ask if such an axiom can be replaced by a "two-dimensional" axiom so that the resulting set of axioms is equivalent to the original one. The only good candidate for such an axiom is the requirement that every plane be an affine plane. (Note that following immediately from the classical axioms of Veblen and Young for projective spaces is the fact that any linear space containing three non-collinear points and having the property that every plane is a projective plane must be a projective space.) in a very beautiful work [12] Buekenhout has shown that this "two-dimensional" axiom can replace * A H the necessary definitions which we take for granted here will be stated in detail in the text. 1 the "three-dimensional" one, provided that the line size is at least four. An HTS has the property that each plane is the nine point affine plane of order 3, while not being isomorphic to the point line incidence structure of some affine geometry over GF(3) • Marshall Hall, Jr. constructed the first such system in [26] in i960. It has eighty-one points and is the unique HTS of minimum size. I should remark that in the literature such systems have sometimes been called affine triple systems; but since they are really only locally affine and since this treatise is by far the most extensive study of them, I have taken the liberty of naming them in Professor Hall's honor. The class of HTS’s in the case of line size three together with the class of Steiner Quadruple Systems in the case of line size two show that Buekenhout's result can not be improved. Also, there is at present a great deal of interest in perfect matroid designs. Quite a number of things have been proved about such geometries. For example, many designs, some of them new in terms of their parameters, can be derived from them. However, the only known examples of non-trivial perfect matroid designs are the classical projective and affine geometries, the t - (v,k,l) designs, and now the HTS's. In 1965 Hall and B. H. Bruck realized that an HTS can be coordinatized by (and, in fact, is equivalent to) a coinnutative Moufang loop of exponent 3* Bruck had developed an extensive theory of loops which in 1973 was exploited to some extent by Peyton Young in [ 67 ] , a paper whi ch provided the impetus for this dissertation.
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