Determining Group Structure from the Sets of Character Degrees
DETERMINING GROUP STRUCTURE FROM THE SETS OF CHARACTER DEGREES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Kamal Aziziheris May 2011 Dissertation written by Kamal Aziziheris B.S., University Of Tabriz, 1999 M.A., University of Tehran, 2001 Ph.D., Kent State University, 2011 Approved by M. L. Lewis, Chair, Doctoral Dissertation Committee Stephen M. Gagola, Jr., Member, Doctoral Dissertation Committee Donald White, Member, Doctoral Dissertation Committee Brett D. Ellman, Member, Doctoral Dissertation Committee Anne Reynolds, Member, Doctoral Dissertation Committee Accepted by Andrew Tonge, Chair, Department of Mathematical Sciences Timothy S. Moerland, Dean, College of Arts and Sciences ii . To the Loving Memory of My Father, Mohammad Aziziheris iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . v INTRODUCTION . 1 1 BACKGROUND RESULTS AND FACTS . 12 2 DIRECT PRODUCTS WHEN cd(Oπ(G)) = f1; n; mg . 22 3 OBTAINING THE CHARACTER DEGREES OF Oπ(G) . 42 4 PROOFS OF THEOREMS D AND E . 53 5 PROOFS OF THEOREMS F AND G . 62 6 EXAMPLES . 70 BIBLIOGRAPHY . 76 iv ACKNOWLEDGEMENTS This dissertation is due to many whom I owe a huge debt of gratitude. I would especially like to thank the following individuals for their support, encouragement, and inspiration along this long and often difficult journey. First and foremost, I offer thanks to my advisor Prof. Mark Lewis. This dissertation would not have been possible without your countless hours of advice and support. Thank you for modeling the actions and behaviors of an accomplished mathematician, an excellent teacher, and a remarkable person.
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