P-GROUP CODEGREE SETS and NILPOTENCE CLASS A
p-GROUP CODEGREE SETS AND NILPOTENCE CLASS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Sarah B. Croome May, 2019 Dissertation written by Sarah B. Croome B.A., University of South Florida, 2013 M.A., Kent State University, 2015 Ph.D., Kent State University, 2019 Approved by Mark L. Lewis , Chair, Doctoral Dissertation Committee Stephen M. Gagola, Jr. , Members, Doctoral Dissertation Committee Donald L. White Robert A. Walker Joanne C. Caniglia Accepted by Andrew M. Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS Table of Contents . iii Acknowledgments . iv 1 Introduction . 1 2 Background . 6 3 Codegrees of Maximal Class p-groups . 19 4 Inclusion of p2 as a Codegree . 28 5 p-groups with Exactly Four Codegrees . 38 Concluding Remarks . 55 References . 55 iii Acknowledgments I would like to thank my advisor, Dr. Mark Lewis, for his assistance and guidance. I would also like to thank my parents for their support throughout the many years of my education. Thanks to all of my friends for their patience. iv CHAPTER 1 Introduction The degrees of the irreducible characters of a finite group G, denoted cd(G), have often been studied for their insight into the structure of groups. All groups in this dissertation are finite p-groups where p is a prime, and for such groups, the degrees of the irreducible characters are always powers of p. Any collection of p-powers that includes 1 can occur as the set of irreducible character degrees for some group [11].
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