ON FINITELY GENERATED NILPOTENT GROUPS AND THEIR SUBGROUPS by BRYAN SANDOR MARTIN EVANS, COMMITTEE CHAIR MARTYN DIXON BRUCE TRACE ALLEN STERN JON CORSON A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of The University of Alabama TUSCALOOSA, ALABAMA 2017 Copyright Bryan Sandor 2017 ALL RIGHTS RESERVED ABSTRACT In this work we investigate nilpotent groups G in which all proper subgroups (or all subgroups of infinite index) have class smaller than the class of G. Our main results are obtained by considering analagous questions for Lie algebras and using the Lazard corre- spondence and the Mal’cev correspondence. Among other things, for each n ≥ 3, we prove the existence of nilpotent groups of class 2n in which every proper subgroup (or subgroup of infinite index) has class at most n. ii DEDICATION I would like to dedicate this work to the memory of Professor Bettina Richmond, my thesis advisor at Western Kentucky University, who started me on my road to success. iii ACKNOWLEDGMENTS Certainly I would never have made it this far without the patient and unconditional guidance of my advisor, Professor Martin Evans. He has the best temperament and per- sonality anyone could hope for in a mentor. I esteem our collaboration and will forever be grateful for his tutelage. My parents, Daniel and Dolly, have been my bedrock of support as I pursued this degree. I do not believe they ever doubted that I could earn this, even when I doubted myself. I lack the vocabulary and elegance of a poet to truly express my love for them. They are my heart. My brother, Adam, is the true artist between us and his passion for his craft served as an inspiration for me. My sister-in-law, Gina, worked very hard for her Doctorate in Veterinary Medicine. Her dedication and determination to settle for nothing less motivated me. My cousins Tom and Kay Bloodworth know the importance of being able to laugh, lending every measure of levity to even the gravest of situations, and there were plenty throughout school that they could always help me laugh through. Their parents, Tay “Scott” and Martha, are spiritual parents to me and were integral to my obedience to the Gospel. My grandfather, William Glenn “Poppy” Bloodworth, is nothing short of the patriarch to our family and has a heart bigger and stronger (despite medical evidence to the contrary) than even the Grinch on Christmas. Halie “Nana” was taken from him and us too soon and I wish she were here to see her eldest grandchild graduate with a doctorate as she was there to see me graduate high school. My best friends, Jen Manchenko and Rachel and Trey Ward, never let me get away with beating myself up and helped me maintain my sanity throughout my studies here. iv My friends Anne Duffee and Erin Watley were wonderful confidantes, helping me realize that it really is normal to feel like I am out of my educational league, even though it is not true. We laughed, complained, cried, and celebrated together throughout the program. I thank the Department of Mathematics at the University of Alabama for giving me financial and professional support, fostering my teaching and researching skills. Finally, I thank G-d, who fashioned me in HIS image and gave me all the skills I possess. Psalm 148 is my favorite to read and to sing but I close with the following: “Let everything that hath breath praise the LORD.” -Psalm 150:6 v CONTENTS ABSTRACT . ii DEDICATION . iii ACKNOWLEDGMENTS . iv LIST OF FIGURES . vii CHAPTER 1. INTRODUCTION . 1 CHAPTER 2. RESULTS MOTIVATED BY THE WORK OF C. K. GUPTA . 11 CHAPTER 3. EXISTENCE THEOREMS . 19 CHAPTER 4. ASSOCIATIVE ALGEBRAS . 34 REFERENCES . 38 vi LIST OF FIGURES 3.6 Young tableau associated with anti-symmetrizer Θ . 24 vii CHAPTER 1 INTRODUCTION In this chapter we collect together various definitions and basic results we will need in the sequel. Definition 1. Let G be a group. The center of G, denoted Z(G), is the subgroup of G consisting of the elements which commute with every element of G. Definition 2. A group G is called nilpotent if it has a central series, that is, a normal series 1 = G0 ≤ G1 ≤ ::: ≤ Gn = G such that Gi+1=Gi is contained in the center of G=Gi for all i. The length of a shortest central series of G is the nilpotent class of G. Definition 3. Let g1; g2; g3; : : : ; gn;::: be elements of a group G. The commutator of g1 −1 −1 and g2 is [g1; g2] = g1 g2 g1g2.A simple commutator of weight n ≥ 2 is inductively defined as [g1; g2; : : : ; gn] = [[g1; g2; : : : ; gn−1]; gn], where [g1] = g1. Recall that the conjugate of g by h is gh := h−1gh for all g; h 2 G. Our first lemma can be proved by direct calculation. Lemma 1.1. Let G be a group and let x; y; z 2 G. Then 1. [x; y]−1 = [y; x], 2. [x; zy] = [x; y][x; z]y and [xz; y] = [x; y]z[z; y], and 3. [x; y−1; z]y[y; z−1; x]z[z; x−1; y]x = 1 (The Hall-Witt identity). Definition 4. Let X be a nonempty subset of a group G. Define the subgroup generated by X, denoted by hXi, to be the intersection of all subgroups of G which contain X. 1 For nonempty subsets X1 and X2 of a group G, let [X1;X2] = h[x1; x2] j x1 2 X1; x2 2 X2i: We similarly define [X1;X2;:::;Xn] inductively by [X1;X2;:::;Xn] = [[X1;X2;:::;Xn−1];Xn]: With this notation we introduce two important central series. Definition 5. The series G = γ1(G) ≥ γ2(G) ≥ γ3(G) ≥ ::: in which γn+1(G) = [γn(G);G] is called the lower central series of G. Notice that every γn(G) is fully-invariant in G and that γn(G)/γn+1(G) lies in the center of G/γn+1(G). Definition 6. The upper central series is given by 1 = ζ0(G) ≤ ζ1(G) ≤ ζ2(G) ≤ :::; where ζn+1(G)/ζn(G) = Z(G/ζn(G)). Each ζn(G) is characteristic but not necessarily fully-invariant in G. Notice that ζ1(G) = Z(G). Lemma 1.2. [16, 5.1.9] Let 1 = G0 ≤ G1 ≤ ::: ≤ Gn = G be a central series in a nilpotent group G. Then (i) γi(G) ≤ Gn−i+1, so that γn+1(G) = 1; 2 (ii) Gi ≤ ζi(G), so that ζn(G) = G; (iii) the nilpotent class of G equals the length of the upper central series, which itself equals the length of the lower central series. In particular, a group G is nilpotent if and only if the lower central series of G reaches the identity subgroup after finitely many steps or, equivalently, the upper central series of G reaches the group itself after finitely many steps. It is worth explicitly mentioning that if G is nilpotent of class c, then γc+1(G) = 1 (but γc(G) 6= 1) and ζc(G) = G (but ζc−1(G) 6= G). Theorem 1.3. Let G be a nilpotent group of class c. Then all of its subgroups are nilpotent and have class at most c. Furthermore, if N ¡ G, then G=N is also nilpotent of class at most c. We now begin a brief discussion of torsion in nilpotent groups. Definition 7. Let π be a nonempty set of prime numbers. A π-number is a positive integer whose prime divisors belong to π. An element of a group G is called a π-element if its order is a π-number. If every element in G is a π-element, then the group is called a π-group. In particular, when π = fpg, we speak of p-elements and p-groups. Note that if G is finite, by Lagrange’s theorem every element is a p-element if and only if the order of G is a power of p. Definition 8. The commutator subgroup of G (or derived subgroup of G), denoted by G0, is the subgroup of G generated by all the commutators. Note that G=G0 is the largest abelian quotient group of G. We call it the abelianization of G and sometimes denote it by Gab. Lemma 1.4. [16, 5.2.6] Let P be a group-theoretical property which is inherited by images of tensor products and by extensions. If G is an nilpotent group such that G=G0 has P, then G has P. 3 For example, let P be the property of being finite. Then we obtain the result: if G is a nilpotent group and G=G0 is finite, then G too is finite. Theorem 1.5. [16, 5.2.7] Let G be a nilpotent group. Then the elements with finite order in G form a fully-invariant subgroup, T (G), called the torsion subgroup of G, such that G=T (G) Q is torsion-free and T (G) = p Tp(G) where Tp(G) is the unique maximum p-subgroup of G. Proof. Let π be a nonempty set of prime numbers and let Tπ(G) denote the subgroup gen- erated by all π-elements of G. Now (Tπ(G))ab is also generated by π-elements and, being abelian, is a π-group. Since G is nilpotent and (Tπ(G))ab is a π-group, the subgroup Tπ(G) is also a π-group.