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CENTRAL CAMINA PAIRS a Dissertation Submitted to Kent State CENTRAL CAMINA PAIRS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by David G. Costanzo May 2020 c Copyright All rights reserved Except for previously published materials Dissertation written by David G. Costanzo B.S., University of Scranton, 2010 M.S., University of Scranton, 2012 M.S., Kent State University, 2018 Ph.D., Kent State University, 2020 Approved by Mark L. Lewis , Chair, Doctoral Dissertation Committee Stephen M. Gagola, Jr. , Members, Doctoral Dissertation Committee Donald L. White Scott A. Courtney Joanne Caniglia Accepted by Andrew M. Tonge , Chair, Department of Mathematical Sciences Mandy Munro-Stasiuk, Ph.D. , Interim Dean, College of Arts and Sciences TABLE OF CONTENTS TABLE OF CONTENTS . iii ACKNOWLEDGMENTS . v 1 INTRODUCTION . 1 2 ELEMENTARY GROUP THEORY . 6 3 CAMINA PAIRS . 12 3.1 Camina Pairs . 12 3.2 Central Camina Pairs . 14 4 MAIN RESULTS ON CENTRAL CAMINA PAIRS . 17 4.1 Central Camina Pairs of Nilpotence Class at Least 4 . 17 4.2 Central Camina Pairs of Nilpotence Class 3 . 20 4.3 Resolving Conjecture (L) in Special Cases . 21 4.4 A Minimal Counterexample to Conjecture (L) . 22 4.5 More Results on Central Camina Pairs . 26 5 A PARTICULAR GRAPH ATTACHED TO A GROUP . 30 5.1 Introduction . 30 5.2 More Elementary Group Theory . 31 5.3 Basic Graph Theory . 32 5.4 Preliminary Observations . 33 5.5 2-Frobenius Groups . 36 BIBLIOGRAPHY . 42 iii DEDICATION This work is dedicated to the memory of my father Gabriel R. Costanzo. iv ACKNOWLEDGMENTS I thank my advisor, Dr. Mark L. Lewis. I thank him for formulating the conjecture that so greatly stimulated my interest. I thank him for his patience as I was presenting trial runs in an attempt to resolve this conjecture. Finally, I thank him for playing such a important role in my mathematical development. Many thanks go to Dr. Donald L. White, whose character theory course in the Spring 2018 semester benefited me greatly. I also thank Dr. White for carefully reading this dissertation and providing me with excellent feedback, which led to many improvements. I thank Dr. Stephen M. Gagola, Jr. for many useful comments on a draft of this dissertation and, most importantly, I thank him for Theorem 6.3 in [3]. I thank Stefano Schmidt, Eyob Tsegaye, and Gabe Udell for an awesome summer of research. I am extremely grateful that our paths crossed. I thank Alexander D. Bongiovanni for many conversations about the graduate student experience. Many thanks go to Edward D. Penetar for years and years of unwavering friendship. I also thank him for his countless attempts to keep me sane. Finally, I thank my mother, Susan A. Costanzo, for her support and patience throughout this process. v CHAPTER 1 INTRODUCTION All groups under consideration are assumed to be of finite order. The primary focus of this dissertation is central Camina pairs. In the last chapter, however, the cyclic graph of a 2-Frobenius group is discussed. Let G be a group. Assume that G has a nontrivial proper subgroup H satisfying H \ Hg = f 1 g for every g 2 G n H. Under this assumption, G is called a Frobenius group and H is called a Frobenius complement. If G is a Frobenius group with Frobenius complement H, then the set ! [ N = G n Hg [ f 1 g g2G turns out to be a normal subgroup of G. The subgroup N is called the Frobenius kernel of G. Let G be a Frobenius group with Frobenius kernel N, and consider the following condition, which is satisfied by the elements in N and the elements in G n N. For every x 2 G n N and for every a 2 N, the element x is conjugate to xa. A.R. Camina [1] studied the structure of a group G with a nontrivial proper normal subgroup N satisfying this condition. One of the main results obtained in [1] is the following classification. Theorem (Camina, [1], Theorem 2). Let G be a group and let N be a nontrivial proper normal subgroup of G. Assume that for every x 2 G n N and for every a 2 N, the element x is conjugate to xa. Then 1 1. G is a Frobenius group with Frobenius kernel N, or 2. G=N or N is a p-group for some prime p. Several authors continued this line of work, and the following definition became standard. If G is a group with a nontrivial proper normal subgroup N such that for every x 2 G n N and for every a 2 N, the element x is conjugate to xa, then (G; N) is called a Camina pair. The notion of a Camina pair has numerous equivalent formulations; see Lemma 3.1 in this dissertation and Lemma 4.1 in [10]. In particular, if N is a nontrivial proper normal subgroup of G, then (G; N) is a Camina pair if and only if for every x 2 G n N and for every a 2 N, there exists an element b 2 G such that a = [b; x]. This condition shall be our official definition of a Camina pair. Let (G; N) be a Camina pair. Then Z(G) ≤ N ≤ G0; see Lemma 3.2. If N = G0, then G is called a Camina group. Camina groups have been the subject of much investigation; see [2], [9], [11], and [12]. The focus in this dissertation is the other extreme: namely, N = Z(G). A Camina pair (G; Z(G)) is called a central Camina pair. Mark L. Lewis pioneered an investigation of central Camina pairs in [8]. Lewis made the following key observation. Theorem (Lewis, [8], Lemma 2.1). If (G; Z(G)) is a Camina pair, then G is a group of prime power order. The proof given in [8] of the previous fact uses Camina's classification result mentioned previously. A proof that does not rely on this classification appears below as Lemma 3.5. The following conjecture was made in [8]. Conjecture (L). If (G; Z(G)) is a Camina pair, then jG : Z(G)j ≥ jZ(G)j2. 2 Lewis was able to obtain the following bound, which appears as Theorem 3 in [8]. Theorem (Lewis, [8], Theorem 3). If (G; Z(G)) is a Camina pair, then jG : Z(G)j > jZ(G)j4=3: Let (G; Z(G)) be a Camina pair, and recall that G must be a p-group for some prime p. Under the additional condition that the factor group G=Z(G) has exponent at least p2, Lewis was indeed able to establish that jG : Z(G)j ≥ jZ(G)j2; in fact, a stronger bound is obtained. The following theorem is Theorem 5 in [8]. Theorem (Lewis, [8], Theorem 5). If (G; Z(G)) is a Camina pair where G is a p-group and the factor group G=Z(G) has exponent pn, n ≥ 2, then jG : Z(G)j ≥ jZ(G)jnpn: Resolving Conjecture (L) is the primary motivation of this dissertation. In contrast to the efforts mentioned above, the approach here is to partition the problem according to the nilpotence class of G. In particular, Conjecture (L) is resolved when the nilpotence class of G is at least 4. Theorem A. If (G; Z(G)) is a Camina pair where the group G has nilpotence class at least 4, then jG : Z(G)j > jZ(G)j2: Caution should be exercised here. If specialized conditions are imposed on a nilpotent group, then there may exist a constant that bounds the nilpotence class. In particular, the condition that (G; Z(G)) forms a Camina pair may force G to have a small nilpotence class, which of course lessen the content of Theorem A. Fortunately, central Camina pairs of nilpotence class at least 4 are abundant. The following theorem recasts Theorem 6.3 in [3] in our present terminology. 3 Theorem (Gagola, [3], Theorem 6.3). If P is a p-group, then there exists a group G such that (G; Z(G)) is a Camina pair and P is isomorphic to a subgroup of G=Z(G). In particular, notice that Gagola's result yields that central Camina pairs can have arbitrarily large nilpotence class. Let (G; Z(G)) be a Camina pair. With Conjecture (L) resolved when G has nilpotence class at least 4, the situation where G has nilpotence class 3 needs to be handled. Unfortunately, Conjecture (L) remains open under this hypothesis. The following inequality, however, is obtained; note that this inequality improves existing bounds. Theorem B. If (G; Z(G)) is a Camina pair where the group G has nilpotence class 3, then jG : Z(G)j > jZ(G)j3=2: Let (G; Z(G)) be a Camina pair and assume that G has nilpotence class 2. Under these hypotheses, G0 = Z(G), and so G is a Camina p-group of nilpotence class 2. I.D. MacDonald [11] proved that jG : Z(G)j ≥ jZ(G)j2 in this situation. Combining this fact with the results above, the following general bound for central Camina pairs is obtained. Corollary C. If (G; Z(G)) is a Camina pair, then jG : Z(G)j > jZ(G)j3=2: 0 Seeking another way to approach Conjecture (L), recall that Z2(G) ≤ CG(G ) for 0 0 any group G. So, if G is a group, then either Z2(G) = CG(G ) or Z2(G) < CG(G ). A strong bound is obtained when (G; Z(G)) is a Camina pair satisfying the condition 0 that Z2(G) < CG(G ).
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