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Maximal Subgroups of Finite Groups MAXIMAL SUBGROUPS OF FINITE GROUPS By LINDSEY-KAY LAUDERDALE 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 2014 c 2014 Lindsey-Kay Lauderdale ⃝ To Mom, Dad and Lisa ACKNOWLEDGMENTS I would like to thank my advisor Dr. Alexandre Turull, who first sparked my interest in group theory. His endless patience and support were invaluable to the completion of this dissertation. I am also grateful for the support, questions and suggestions of my committee members Dr. Kevin Keating, Dr. David Mazyck, Dr. Paul Robinson and Dr. Peter Sin. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 LIST OF TABLES ..................................... 6 ABSTRACT ........................................ 7 CHAPTER 1 DISSERTATION OUTLINE ............................. 9 2 A BRIEF BACKGROUND OF FINITE GROUP THEORY ........... 11 2.1 Introduction ................................... 11 2.2 Maximal Subgroups ............................... 14 2.3 Notation and Some Definitions ......................... 17 2.4 On the Number of Maximal Subgroups .................... 18 3 UPPER BOUNDS FOR THE NUMBER OF MAXIMAL SUBGROUPS .... 20 3.1 Introduction ................................... 20 3.2 Upper Bounds Proven by Others ....................... 21 3.3 Preliminary Results ............................... 22 3.4 Proof of Main Theorem ............................. 23 4 LOWER BOUNDS FOR THE NUMBER OF MAXIMAL SUBGROUPS .... 25 4.1 Introduction ................................... 25 4.2 Preliminary Results ............................... 25 4.3 Proof of Main Theorem ............................. 29 4.4 A Special Case ................................. 35 5 SOME FURTHER LOWER BOUNDS FOR THE NUMBER OF MAXIMAL SUBGROUPS ..................................... 38 5.1 Introduction ................................... 38 5.2 Definitions .................................... 40 5.3 Preliminary Results ............................... 41 5.4 m(G) for p-Solvable Groups ......................... 43 5.5 |m(G)| for Nonsolvable Groups ........................ 46 5.6 Proof| | of Main Theorems ............................ 51 5.7 Remarks on α(G) ................................ 52 5.8 Remarks on Property B ............................ 54 REFERENCES ....................................... 56 BIOGRAPHICAL SKETCH ................................ 57 5 LIST OF TABLES Table page 5-2 Values of α(G). .................................... 54 6 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MAXIMAL SUBGROUPS OF FINITE GROUPS By Lindsey-Kay Lauderdale May 2014 Chair: Alexandre Turull Major: Mathematics Finite group theory is a topic that has held the attention of mathematicians for over ahundredyears.Thisisinpartduetoitsapplicationsthroughoutmultiplebranchesof science including biology, chemistry and physics. Investigating finite groups is a classical problem, with many open questions remaining. A major component to understanding finite groups is comprehending their structure. One way to investigate the structure of finite groups is to study their maximal subgroups. In this dissertation, we study the number of maximal subgroups in a finite group G,denoted m(G) . When G is a cyclic group, an elementary calculation proves that | | m(G) = π(G) ,whereπ(G)denotesthesetofprimeswhichdivide G .Wefurtherprove | | | | | | that if G = P P P ,then m(G) = m(P ) + m(P ) + + m(P ) , where 1 × 2 ×···× n | | | 1 | | 2 | ··· | n | π(G)= p ,p ,...,p and P Syl (G)foreachi 1, 2,...,n . { 1 2 n} i ∈ pi ∈{ } We proceed by turning our attention to bounding the number of maximal subgroups in an arbitrary finite group. First, we consider the upper bound for the number of maximal subgroups in a finite group. Upper bounds for the number of maximal subgroups have gradually been sharpened by other mathematicians through making further and further assumptions on the group. We make more assumptions on the structure of a finite group and then improve the existing upper bound for the number of the maximal subgroups in a finite solvable group. 7 To continue to narrow the range of the possible number of maximal subgroups in a finite group, we must consider the lower bound. We prove multiple lower bounds for the number of maximal subgroups in an arbitrary noncyclic finite group. In general, for a noncyclic group G, m(G) π(G) + p where p π(G)isthesmallestprimethatdivides | |≥| | ∈ G .IfG has a noncyclic Sylow subgroup and q π(G)isthesmallestprimesuchthat | | ∈ Q Syl (G)isnoncyclic,then m(G) π(G) + q.Weconcludebyproducingtwonew ∈ q | |≥| | lower bounds for m(G) ,bothofwhichconsideralloftheprimesinπ(G). | | 8 CHAPTER 1 DISSERTATION OUTLINE This first chapter gives an outline of the remaining chapters in this dissertation. The main content begins in Chapter 2 with some basic definitions and a natural way of classifying groups by their structural properties. We proceed with some results which provide a foundation for the study of maximal subgroups of finite groups, the focus of this dissertation. After a brief introduction to maximal subgroups, we calculate the number of maximal subgroups of some basic finite groups. However, with minimal assumptions on a group we cannot always calculate the exact number of maximal subgroups. Thus we attempt to narrow the range for the possible number of maximal subgroups in a finite group by considering both upper and lower bounds. Chapter 3 focuses on the upper bounds for the number of maximal subgroups. Here we only consider the number of maximal subgroups in finite solvable groups. Several upper bounds for the number of maximal subgroups have been calculated by other mathematicians, each one improving the upper bound that was proven before. In the last section of Chapter 3,wemakesomefurtherassumptionsonastructureofafinitesolvable group to improve the upper bound for the number of the maximal subgroups. To continue to narrow the range for the possible number of maximal subgroups of agivenfinitegroup,wenextconsiderthelowerbound.BothChapter4 and Chapter 5 consider the lower bounds for the number of maximal subgroups in a finite group. In 2013, the main content of Chapter 4 appeared in Volume 101 of Archiv der Mathematik and was titled, ‘Lower bounds on the number of maximal subgroups in a finite group’ (see [6]). This chapter focuses on producing a lower bound for the number of maximal subgroups in afinitegroupandthenproceedstoimprovetheboundbymakingminimalassumptionson the structure of the group. We conclude this chapter by providing an example of a group which achieves the stated lower bounds. 9 In Chapter 5,weimprovethepreviouslystatedlowerboundsforthenumberof maximal subgroups by making further assumptions on the structure of the group. In particular, the lower bound is improved by partitioning the set of primes which divide the order of the group into three sets. Again, we conclude the chapter by providing examples of groups which achieve the stated lower bounds. The main content of Chapter 5 is currently submitted. 10 CHAPTER 2 A BRIEF BACKGROUND OF FINITE GROUP THEORY 2.1 Introduction Groups are mathematical objects which measure symmetry. They are studied throughout many areas of mathematics, as well as multiple branches of science. A group is defined below: Definition 2.1.1. Agroupisanorderedpair(G, ), where G is a set and is a binary ∗ ∗ operation on G satisfying the following axioms: (i) (a b) c = a (b c), for all a, b, c G ∗ ∗ ∗ ∗ ∈ (ii) there exists an element e G such that for all a G we have a e = e a = a ∈ ∈ ∗ ∗ 1 1 1 (iii) for each a G there is an element a− of G such that a a− = a− a = e. ∈ ∗ ∗ This simple definition of a group can yield extremely difficult questions which arise in group theory. In this dissertation, we study finite groups only and part of finite group theory attempts to classify finite groups. One of the simplest ways of classifying finite groups is by their order, or the number of elements in the group. A list of small groups, classified by order, can be found in the program Groups, Algorithms, and Programming (GAP) [4]. A more natural way to classify groups is by their structural properties. These structural properties form a hierarchy, as seen below: Cyclic Groups Abelian Groups Nilpotent Groups Solvable Groups All Groups ⊂ ⊂ ⊂ ⊂ The simplest structural property is seen in cyclic groups. A cyclic group is a group which is generated by a single element. Every finite cyclic group is isomorphic to Zn, where n is a positive integer. All cyclic groups are abelian groups, that is for all a, b G ∈ we have a b = b a. Abelian groups are completely classified and the classification can be ∗ ∗ seen in the following theorem: Theorem 2.1.2. Let G be a finite abelian group. Then G = Zn1 Zn2 Znm , ∼ × ×···× 11 for some integers n1,n2,...,nm satisfying the following conditions: (1) (i) n 2 for all i 1, 2,...,m i ≥ ∈{ } (ii) n n for j 1, 2,...,m 1 j+1| j ∈{ − } (2) The expression in (1) is unique, up to isomorphism. Proof. See for example [3,Theorem3.22]. The next category of groups is nilpotent groups. Nilpotent groups enjoy the following equivalent properties: Theorem 2.1.3. Let G be a finite group. Suppose p1,p2,...,pn are the distinct primes dividing the order of G and let P be a Sylow p -subgroup of G for i 1, 2,...,n . Then i i ∈{ } the following are equivalent: (1) G is nilpotent (2) G = P P P ∼ 1 × 2 ×···× n (3) For all i 1, 2,...,n , P
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