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The Characters and Commutators of Finite Groups

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The Characters and Commutators of Finite Groups THE CHARACTERS AND COMMUTATORS OF FINITE GROUPS By TIM W. BONNER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 °c 2009 Tim W. Bonner 2 ACKNOWLEDGMENTS I am sincerely grateful to my adviser, Alexandre Turull. With unyielding patience and constant support, he has been integral in my academic and personal growth. I feel indebted to have been his student, and I am fortunate to know such an inspiring individual. I also will forever appreciate the encouragement of my family. Over the past six years, my parents and sister have responded to each step forward and every setback with calm assurance. Finally, my wife, Emily, has witnessed it all only a glance away, and this work undeniably bears the steady strength of her hand. 3 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 3 LIST OF TABLES ..................................... 5 ABSTRACT ........................................ 6 CHAPTER 1 INTRODUCTION .................................. 8 1.1 Products of Commutators ........................... 8 1.2 Further Generation Problems ......................... 9 1.2.1 Products of Conjugacy Classes ..................... 9 1.2.2 Products of Characters ......................... 10 1.3 The Taketa Problem .............................. 11 2 MATHEMATICAL PRELIMINARIES FOR CHAPTERS 3 AND 4 ....... 13 3 PRODUCTS OF COMMUTATORS AND BARDAKOV'S CONJECTURE ... 17 3.1 A Character Identity of Burnside ....................... 18 3.2 Analysis and Results .............................. 21 3.3 Bardakov's Conjecture ............................. 26 4 PRODUCTS OF CONJUGACY CLASSES AND CHARACTERS ........ 28 4.1 Conjugacy Class Covering Numbers ...................... 28 4.2 Comparison with Previous Bounds ...................... 33 4.3 Character Covering Numbers ......................... 36 4.4 Comparison with Previous Bounds ...................... 39 5 MATHEMATICAL PRELIMINARIES FOR CHAPTER 6 ............ 43 6 THE TAKETA PROBLEM ............................. 48 6.1 Derived Length vs. Number of Character Degrees in Certain p-groups ... 48 6.2 Introduction ................................... 48 6.3 Normally Serially Monomial p-groups ..................... 49 REFERENCES ....................................... 64 BIOGRAPHICAL SKETCH ................................ 67 4 LIST OF TABLES Table page 4-1 Bound Comparison for cn(C) for A5 ........................ 34 4-2 Bound Comparison for cn(C) for SL2(5) ...................... 35 4-3 Bound Comparison for cn(C) for SL2(8) ...................... 35 4-4 Bound Comparison for cn(C) for Perfect Group of Order 1080 .......... 35 4-5 Bound Comparison for cn(C) for M11 ........................ 36 4-6 Bound Comparison for ccn(Â) for A5 ........................ 40 4-7 Bound Comparison for ccn(Â) for SL2(8) ...................... 41 4-8 Bound Comparison for ccn(Â) for P SU3(3) ..................... 41 4-9 Bound Comparison for ccn(Â) for M11 ....................... 41 5 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy THE CHARACTERS AND COMMUTATORS OF FINITE GROUPS By Tim W. Bonner August 2009 Chair: Alexandre Turull Major: Mathematics Let G be a ¯nite group. It is well-known that the elements of the commutator subgroup must be products of commutators, but need not themselves be commutators. A natural question is to determine the minimal integer, ¸(G), such that each element of the commutator subgroup may be represented as a product of ¸(G) commutators. An analysis of a known character identity allows us to improve the existing lower bounds for jGj in terms of ¸(G). The techniques we develop also address the related following question. Suppose we have a conjugacy class C of a ¯nite group G such that hCi = G = G0. One may ask for the minimal integer cn(C) such that each element of G may be expressed as a product of cn(C) elements of the conjugacy class. Again, we improve the known upper bounds, this time for cn(C). Our second focus is the relation between the derived length of a ¯nite solvable group and the cardinality of the set of character degrees in the same group. Over the past few decades, this topic has been explored by Isaacs, Gluck, Slattery, and most recently, by Thomas Keller. There is a standing conjecture that universal constants C1 and C2 exist such that for any ¯nite solvable group G, dl(G) · C1 log jcd(G)j + C2: Indeed, Thomas Keller has reduced the conjecture to the case of p-groups, and proceeded to attack this case by a study of normally monomial p-groups of maximal class. We extend 6 and re¯ne his methods to a broader class of groups, those for which each irreducible character may be induced from a single normal series. We also examine the special properties held by these groups, said to be normally serially monomial. 7 CHAPTER 1 INTRODUCTION 1.1 Products of Commutators The commutator structure of a ¯nite group, G, has been an object of consistent study since the end of the 19th century. It has been noted by Frobenius [13], that Dedekind was the ¯rst to introduce the idea of a commutator, an element of the group of the form a¡1b¡1ab, for a; b 2 G. Dedekind also initiated the study of the subgroup generated by the set of commutators, later to be called the commutator subgroup. It was soon recognized that each element of the commutator subgroup need not be a commutator, and such an element is said to be a noncommutator. William Benjamin Fite [12] was the ¯rst to publish an example of a group containing such an element, though he attributed the example to G.A. Miller. William Burnside [5], in 1903, subsequently developed a criterion to determine whether an element of the commutator subgroup was indeed a commutator. He showed that g 2 G was a commutator if and only if X  (g) 6= 0; (1{1)  (1) Â2Irr(G) where Irr(G) denotes the set of irreducible complex characters of G. This identity can be a powerful tool in determining the existence of noncommutators and we shall make signi¯cant use of a generalization of (1{1). The question then arises as to identifying of the minimal integer, ¸(G), such that every element of the commutator subgroup may be written as a product of ¸(G) commutators. This invariant ¸(G) has been a source of consistent investigation throughout the 20th and 21st centuries. In the 1960's Patrick Gallagher [15] determined an inequality between the size of ¸(G) and the order of a ¯nite group, jGj. Later, in 1982, Robert Guralnick [18] demonstrated that for any positive integer n, one may construct a ¯nite group G such that ¸(G) = n. He also determined the minimal ¯nite groups G, with respect to order, such that ¸(G) 6= 1 [19]. The famous conjecture of Oystein Ore [38], stating that ¸(G) = 1 for any ¯nite simple 8 group G, has been the object of active research through 2008. Contributors to the Ore conjecture include Gow [17], Ellers and Gordeev [11], and Liebeck, O'Brien, Shalev, and Tiep [32]. Our work regarding ¸(G) returns to the earlier considerations of Gallagher. We improve the known lower bound for jGj for a given value of ¸(G) and obtain the following as our ¯rst main result. Theorem A. For any ¯nite nonabelian group G, we have jGj ¸ (¸(G) + 1)! (¸(G) ¡ 1)!: This improves a similar inequality of Gallagher [15] by a factor of 2. Moreover, we use this result to con¯rm and strengthen a conjecture of V.G. Bardakov (3.0.18) posed in the most recent edition of the Kourovka Notebook [36]. Precisely, we obtain the following theorem. Theorem 1.1.1. Let G be a ¯nite nonabeliam group. Then, provided jGj ¸ 1000, we have, ¸(G) 1 · : jGj 250 We remark that Kappe and Morse [23] have shown that ¸(G) 2 f1; 2g for all groups, G, such that jGj · 1000. Our results regarding products of commutators have been published in a 2008 volume of the Journal of Algebra [4]. 1.2 Further Generation Problems 1.2.1 Products of Conjugacy Classes n Let C be a conjugacy class of a ¯nite group G. De¯ning C = fg1 : : : gn j gi 2 Cg, one may ask the following questions. Does there exist a value of n such that Cn = G, and if so, what upper bounds can be placed on the smallest such integer? A very famous related conjecture is that attributed to John Thompson, who suggested that for any ¯nite simple group G, one may always ¯nd a conjugacy class, C such that C2 = G. Indeed, one may show that this implies the Ore conjecture stated in the ¯rst section. Returning to our discussion, it was determined by Arad, Stavi, and Herzog [1] in 1985 that the existence of an integer n such that Cn = G was equivalent to the conjugacy class generating G, 9 i.e. hCi = G, and G being perfect. In the case that G is simple, very strong bounds have been obtained, see for example the work of Liebeck and Shalev [33]. We will consider the general case of G perfect, with the minimal integer called the conjugacy class covering number and denoted cn(C). In their work, Arad, Stavi, and Herzog [1] obtained a variety of upper bounds on the conjugacy class covering number. Towards the latter end of the 1990's and into the ¯rst part of the current decade, David Chillag ([9], [10]) produced alternate upper bounds for cn(C). In many cases, we improve the known upper bounds of the conjugacy class covering number, our main result being the theorem below. Theorem B. Let G be a ¯nite perfect group and C a conjugacy class of G such that n hCi = G. We de¯ne e1(C) = min fn 2 Z+ j 1 2 C g and for  2 Irr(G), we take Â(C) to be the value of  on an element of the conjugacy class C.
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