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Connections Between the Number of Constituents and the Derived Length of a Group a Dissertation Submitted to Kent State Universi Connections Between the Number of Constituents and the Derived Length of a Group A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by, Lisa Rose Hendrixson May, 2017 Dissertation written by Lisa Rose Hendrixson B.S., Kent State University, 2011 M.A., Kent State University, 2013 Ph.D., Kent State University, 2017 Approved by Mark L. Lewis , Chair, Doctoral Dissertation Committee Stephen M. Gagola , Members, Doctoral Dissertation Donald L. White , Committee Joanne Caniglia , Hassan Peyravi , Accepted by Andrew Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences Table of Contents Dedication iv Acknowledgments v 1 Introduction 1 2 Background 5 2.1 Group Theory . .5 2.2 Character Theory . .9 3 Known Results 13 4 Two Nonprincipal Irreducible Constituents 22 5 Three Nonprincipal Irreducible Constituents-The Special Case 33 6 Three Nonprincipal Irreducible Constituents-The General Case 58 7 Examples 85 Bibliography 93 iii To my mother, for her continued love and kindness, and to my father, who gave me a love of mathematics. Acknowledgments I would like to thank my advisor, Dr. Mark Lewis, for his assistance and support throughout my education. I would like to thank Dr. White and Dr. Gagola for their many helpful suggestions that have improved this document and the works and talks that came out of it. Lastly, I would like to thank my parents for their unwavering support and understanding. v Chapter 1 Introduction In this dissertation, we discuss the relationship between the structure of a finite solvable group and the number of nonprincipal irreducible constituents belonging to a particular product of characters. Our main focus is to answer, at least in part, the following question: Is the derived length bounded in terms of the number of nonprincipal irreducible constituents of a particular product of characters, namely χχ? Both Adan-Bante and Keller have made strides in answering this question. In [1], Adan-Bante proves that G= ker(χ) has derived length bounded by a linear function involving the number of nonprincipal irreducible constituents of χχ. Also, Keller was able to prove in [12] that G=F2(G) is bounded in terms of the number of character degrees, although he notes that the bound proved in that paper is not the best possible and suggests what it should be. The notation F2(G) will be defined later. In [2], Adan-Bante completely classifies the solvable groups which have a faithful irreducible character χ such that χχ has one nonprincipal irreducible constituent, and gives some group structure of those solvable groups having an irreducible faithful character χ such that χχ has two nonprincipal irreducible constituents. We begin 1 by proving that the best possible bound when χχ has two nonprincipal irreducible constituents is 8. In fact, we are able to prove that the two constituents in question must be real characters. Then we consider the situation when χχ has three such constituents. This includes the special case where two of the nonprincipal irreducible constituents are complex conjugates, and the general case where none of them are related. Theorem A. Let G be a finite solvable group and let χ 2 Irr(G) be a faithful char- acter. Assume that χχ = 1G + m1α1 + m2α2; # where α1; α2 2 Irr(G) are distinct characters and m1 and m2 are strictly positive integers. Then dl(G) ≤ 8. Throughout, G is a finite solvable group with center Z(G), and the set of irre- ducible characters of G is denoted by Irr(G). Also, for a group V , we denote the set of nonidentity elements of V by V #. Theorem B. Let G be a finite solvable group with a faithful character χ 2 Irr(G) such that χχ = 1G + m1α1 + m2α2; # where α1; α2 2 Irr(G) are distinct characters and m1; m2 are strictly positive inte- gers. Then both α1 and α2 are real-valued characters. The next natural question to consider is the situation where χχ has three non- principal irreducible constituents. This will be broken into two cases. The first to 2 be considered will be when χχ has complex-valued irreducible constituents, which is handled by Theorem C. When we are finished with Theorem C, we will turn our attention to Theorem D, the case when χχ has only real nonprincipal irreducible constituents. We will also see that Theorem C is a consequence of Theorem B. Theorem C. Let G be a finite solvable group with center Z = Z(G) and let χ 2 Irr(G) be a faithful character. Assume that χχ = 1G + m1α1 + m2α2 + m2α2; # where α1; α2 2 Irr(G) are distinct characters and m1 and m2 are strictly positive integers. Then, 1. the order of G is even, 2. dl(G) ≤ 6, 3. both ker(α1) and ker(α2) are abelian groups with either ker(α1) = Z or ker(α2) = Z, and χ(1) is a power of a prime. Theorem D. Let G be a finite solvable group and let χ 2 Irr(G) be a faithful char- acter. Assume that χχ = 1G + m1α1 + m2α2 + m3α3; # where αi 2 Irr(G) are distinct characters and mi are strictly positive integers for all i = 1; 2; 3. Then dl(G) ≤ 16. 3 It should be mentioned that the largest example we have has derived length 6, and is presented at the end of this paper. This suggests that there is some improvement that could be made to the bound in Theorem D, as well as a possibly larger example that has yet to be found. It seems likely that both of these could be true. The next theorem provides a starting point when looking for more examples. In [1], Adan-Bante only proves an implicit bound for the derived length of groups with the desired property. Therefore, it seems likely that the following theorem could also provide some information on finding an explicit bound for the groups in question. Theorem E. Let G be a finite group with E ≤ G a nonabelian subgroup. Assume E=Z is an abelian chief factor of G with Z = Z(G), and that G=E acts with n ≥ 2 # orbits on (E=Z) . Let χ 2 Irr(G) be faithful and suppose that χE 2 Irr(E). Then n X χχ = 1G + αi; i=1 # where αi 2 Irr(G) are distinct and ker(αi) = Z for all i, 1 ≤ i ≤ n. 4 Chapter 2 Background 2.1 Group Theory Since our discussion will center on groups, it is necessary to begin by introducing those topics that will play an important role. The references that will be used are [10] and [11]. A group is a set G together with an associative binary operation ◦ defined on G such that there exists e 2 G with the properties that for each x 2 G, x ◦ e = e ◦ x = x and for each x 2 G there exists some y 2 G such that x ◦ y = y ◦ x = e. If the binary operation is also commutative, then the group is called an abelian group. A group is said to be cyclic if there exists some g 2 G with G = fgnjn 2 Zg; these groups are automatically abelian. We say that H is a subgroup of G if H is a subset of G which is also a group. A proper subgroup H is called maximal if whenever H ≤ K ≤ G then either H = K or K = G. The Frattini subgroup, denoted Φ(G), is the intersection of all maximal subgroups. Also, we denoted the index of H in G by jG : Hj, which can be thought of as jGj=jHj when the groups under consideration are finite. As all of the groups in this paper will be finite, this is appropriate. There are several particular types of subgroups which will play a role in our 5 discussion. The first is the normal subgroup. A subgroup N ≤ G is called normal when N g = g−1Ng = N for all g 2 G, and is denoted by N/G. The socle of G is the subgroup of G generated by all of the minimal normal subgroups of G. The centralizer of X ⊆ G is denoted CG(X) and is the set of elements in G which commute with all elements of X. Likewise, the center of G is the set of elements which commute with all elements of G, and is denoted by Z(G). Also relevant to our topics is the commutator subgroup. The commutator of two elements x; y 2 G is denoted by [x; y] = x−1y−1xy and the commutator subgroup is the subgroup G0 = [G; G]. This subgroup is generated by all commutators of two elements in G, i.e., G0 is the set of all finite products of elements of the form [x; y] for x; y 2 G. Let H be a subgroup of G. Then G=H is the set fgHjg 2 Gg. If H is a normal subgroup of G, then G=H is referred to as the factor group, or quotient group. In particular, a chief factor L=K of a group G is a factor group, where K < L are normal subgroups of G and there are no normal subgroups M such that K < M < L. A subgroup P of G is called a p-group if the order jP j is equal to a power of some prime p. This group is also elementary abelian if every nonidentity element of P has the same prime order p and P is an abelian group.
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