C 2010 KENDALL NICOLE MAURER ALL RIGHTS RESERVED

c 2010 KENDALL NICOLE MAURER ALL RIGHTS RESERVED MINIMALLY SIMPLE GROUPS AND BURNSIDE’S THEOREM A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Kendall Nicole Maurer May, 2010 MINIMALLY SIMPLE GROUPS AND BURNSIDE’S THEOREM Kendall Nicole Maurer Thesis Approved: Accepted: Advisor Dean of the College Dr. James P. Cossey Dr. Chand K. Midha Faculty Reader Dean of the Graduate School Dr. Jeffrey Riedl Dr. George R. Newkome Faculty Reader Date Dr. Antonio Quesada Department Chair Dr. Joseph Wilder ii ABSTRACT William Burnside’s paqb theorem is a very important result in group theory, which states that any group G of order paqb is solvable. An interesting fact about this theorem is that it was originally proven using techniques from character theory, another branch of algebra. In fact, it was about seventy years before a group-theoretic proof of Burnside’s theorem was developed through the work of Goldschmidt, Matsuyama, Bender, and other mathematicians. Their approach to proving the theorem was to show that, in essence, minimally simple groups of size paqb do not exist. Our purpose here is to use various techniques from the group-theoretic proof of Burnside’s theorem to establish and prove similar results about minimally simple groups G of arbitrary order. iii ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my wonderful family and friends, especially my mother and father; my sisters, Michelle and Staci; and Grandma Mikey. You always inspire me and give so much of yourselves. Without your continuous love, support, and encouragement this thesis would not have been possible. I also want to express a special thank you to the dedicated faculty and staff of The University of Akron for helping to make my time at the university such a rewarding and enlightening educational experience. I cannot thank Dr. Riedl and Dr. Quesada enough for their time and effort spent reviewing my thesis. Most im- portantly, I would like to recognize my advisor, Dr. Cossey, for first introducing me to the wonders of group theory and for his guidance and patience throughout this process. To all of you, it was a true honor to have your valuable input and was such a privilege to share in your knowledge of mathematics. iv TABLE OF CONTENTS Page LISTOFTABLES................................. vii CHAPTER I. INTRODUCTION............................... 1 1.1 HistoricalBackground .......................... 1 1.2 PreliminaryDefinitions . 2 1.3 MainResults ............................... 4 II. DEFINITIONS AND THEOREMS . 6 2.1 Sylow Subgroups, Hall Subgroups, and Related Theorems . 6 2.2 SolvableGroupsandNilpotentGroups . 10 2.3 Special Types of Subgroups and Elements . 11 2.4 OtherNecessaryTheorems . 12 III. OUTLINE OF THE GROUP-THEORETIC PROOF OF BURN- SIDE’STHEOREM.............................. 13 3.1 PreliminarySteps ............................ 13 3.2 AnalysisofMaximalSubgroups: Steps1-3 . 14 3.3 CentralElements:Steps4-6. 16 v 3.4 G HasOddOrder:Step7........................ 17 3.5 WeObtainaContradiction: Steps8-9 . 18 IV.RESULTS ................................... 19 4.1 Preliminary Results and Statements of Our Main Results . 19 4.2 Results That Do Not Require p-Goodness ............... 24 4.3 FinalMainResults............................ 29 V. GAPRESULTS................................ 34 5.1 Method .................................. 34 5.2 DescriptionofTables. 35 5.3 Tables................................... 37 5.4 PossibleFutureResearch . 40 BIBLIOGRAPHY ................................. 41 vi LIST OF TABLES Table Page 5.1 Groups of the Type PSL(2, 2p) ...................... 37 5.2 Groups of the Type PSL(2, 3p) and PSL(2,p).............. 38 5.3 PSL(2, 23), Sz(8), and PSL(3, 3)..................... 39 vii CHAPTER I INTRODUCTION 1.1 Historical Background In 1904, William Burnside proved his paqb-theorem, which is considered a major result in group theory and states: Theorem 1.1 (Burnside’s Theorem). Let G be a finite group of order paqb, where p and q are primes and a, b ≥ 1. Then G is solvable. Recall that intuitively, a group is called solvable if it can be broken down into smaller pieces as a result of having many normal subgroups. (We will more formally define solvability later in this chapter.) It would seem that such an important result would have a proof that uses very difficult techniques. However, Isaacs [1] considers Burnside’s proof from 1904 [2] to be “elegant and not very difficult.” Also, it is interesting to note that although Burnside’s theorem is a result from group theory, the first proof of the theorem relied on techniques from character theory. This fact causes one to wonder if it is possible to prove Burnside’s Theorem without the use of character theory. Indeed it is, and in the 1970s, efforts were made to develop a proof that only relied on group-theoretic techniques. According to Isaacs 1 [1], one person in particular, David Goldschmidt, used some of John Thompson’s techniques to prove the case of Burnside’s theorem where both p and q are odd [3]. Later, both Helmut Bender [4] and Hiroshi Matsuyama [5] proved the case where p = 2. Thus, about seventy years after the character theory proof of Burnside’s theorem was published, a group-theoretic proof had finally been developed. With this situation in mind, one cannot help but wonder if the group-theoretic techniques used to prove Burnside’s theorem could be used to develop results for groups whose order is not restricted by the paqb requirement. Indeed, the major aim of this paper is to establish results along those lines. In order to obtain a contradiction, the group-theoretic proof of Burnside’s theorem focuses on the analysis of a minimal counterexample to the theorem. This minimal counterexample is an example of what Thompson classified as a minimally simple group [6], which we will define later in this chapter. Thus, it should be noted that the group-theoretic proof of Burnside’s theorem shows that there does not exist a minimally simple group of order paqb. Since our results were developed by using some of these group-theoretic techniques, our results are also concerned with minimally simple groups, rather than an arbitrary group G. Thus, before we are able to state our results, a few important terms should be defined. 1.2 Preliminary Definitions Definition 1.2. A simple group is a group that does not have any proper normal subgroups other than the trivial subgroup. 2 Thus, we can think of a simple group as being a “building block” group that cannot be broken down any further with regards to normal subgroups. Definition 1.3. A group G is solvable if there exist normal subgroups Ni ⊳ G such that 1= N0 ⊆ N1 ⊆ N2 ⊆···⊆ Nr = G, where Ni/Ni−1 is abelian for 1 ≤ i ≤ r. While the above definiton appears technical, we can intuitively understand solvability as a condition that allows us to figuratively “pull apart” a group in order to analyze its components. Definition 1.4. A group G is minimally simple if G is simple and, for each subgroup H < G, H is solvable. The following is a necessary, though nonstandard, definition. Definition 1.5. Let p be a prime dividing the order of a group G. We say a group G is p-good if, for each maximal subgroup M of G, we have Op′ (M) is nilpotent. If π is a set of primes which divide |G|, we say G is π-good if G is p-good for each prime p ∈ π. We will formally define nilpotence and Op′ (M) in Chapter 2. In the meantime, we can think of nilpotence as being a condition similar to solvability that allows us to “break” a group into pieces. In fact, all nilpotent groups are also solvable groups. Definition 1.6. Let π be a set of primes. A Hall π-subgroup of a finite group G is a π-subgroup with index involving no prime of π. 3 Definition 1.7. Let G be a group and let M be a subgroup of G. Then the set of bottom primes of M, b(M), is defined to be {p prime: Op(M) is nontrivial}. The set of top primes of M, t(M), is defined to be {p prime: p | |G : M|}. 1.3 Main Results We are now prepared to state our main results, which concern the sets b(M) and t(M) in the case where M is a maximal subgroup of a minimally simple group G. These results were established (using the methods from the group-theoretic proof of Burnside’s theorem) in the search for a proof or a counterexample to the following conjecture. Conjecture 1.8. Let G be a minimally simple group. Then for every maximal subgroup M of G, we have |b(M) ∩ t(M)| ≤ 1. To put the above conjecture into perspective, it is useful to note that one of the main steps in the group-theoretic proof of Burnside’s theorem found in [1] involves showing that in any minimal counterexample group G to Burnside’s theorem, b(M) ∩ t(M)= ∅ for all maximal subgroups M of G. The following three theorems are our main results. It should be noted that while we would prefer to eliminate the p-goodness hypothesis for Theorems 1.10 and 1.11, we are unable to do so at this time. Also, by using the computer program GAP to analyze some of the minimally simple groups (which were all classified by Thompson in [6]), we have shown that Theorem 1.10 is the strongest possible result of 4 its type, because we have found an example where |b(M) ∩ t(M)| = 1. Additionally, we have found an example of a minimally simple group that is not p-good for some prime p | |G|.

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