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 the- orem 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 sub- group 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|. Therefore, Conjecture 1.8 and Theorem 1.10 below have distinct hypotheses. Our GAP results can be found in Chapter 5.
Theorem 1.9. If G is minimally simple and M is a maximal subgroup of G, with
|G : M| = qn, where q is a prime, then b(M) ∩ t(M)= ∅.
Theorem 1.10. Let G be a minimally simple group that is p-good for all primes p dividing |G|, and let M be a maximal subgroup of G. Then |b(M) ∩ t(M)| ≤ 1.
Theorem 1.11. Let G be a minimally simple group that is p-good for all primes p dividing |G|, and let M be a maximal subgroup of G. Suppose b(M) ∩ t(M) = {p}.
Let π = b(M) \ {p}. Then G has a Hall π-subgroup contained in M.
Intuitively, Theorem 1.9 is a special case of Theorem 1.10, and Theorem 1.11 begins to analyze the case when |b(M) ∩ t(M)| = 1.
The importance of our results is due to the fact that the techniques used in our proofs may someday contribute to proofs of other open problems in group theory.
Furthermore, it is our specific hope that the work we have done here will lead to either a proof or a counterexample to the conjecture that for every maximal subgroup M of a minimally simple group G, we have |b(M) ∩ t(M)| ≤ 1.
5 CHAPTER II
DEFINITIONS AND THEOREMS
The following definitions and theorems are useful for understanding the statements and proofs of our results.
2.1 Sylow Subgroups, Hall Subgroups, and Related Theorems
The first definition concerns various types of groups that can be classified according to the primes dividing their order.
Definition 2.1. Suppose p is a prime. We say a group G is a p-group if the order of G is a power of p. On the other hand, we say a group G is a p′-group if the order of G is not divisible by p. Similarly, if π is a set of primes, we say a group G is a
π-group if the order of G is only divisible by primes found in π, and we say a group
G is a π′-group if the order of G is not divisible by any primes found in π.
The following definition concerns more general usage of the “dash” notation.
Definition 2.2. Let G be a group and let p be a prime. Then we define p′ as the set of all primes that are not equal to p. Similarly, if π is a set of primes, we define π′ as the set of all primes that are not in π.
6 An important property of nontrivial p-groups, which will be used multiple times in the proofs of our results, follows.
Theorem 2.3. Let G be a nontrivial p-group. Then G has a nontrivial center.
This leads us to the definition of a special type of p-subgroup of a group G, called a Sylow p-subgroup of G, which is important in the proofs of our results.
Definition 2.4. Suppose G is a group whose order can be written as |G| = pam where p is a prime and p ∤ m. A Sylow p-subgroup of G is a subgroup of G whose order equals pa.
Some interesting and fundamental results concerning Sylow p-subgroups, called the Sylow E, Sylow C, and Sylow D theorems, follow. They will be used a number of times in the proofs of our results.
Theorem 2.5 (Sylow E). Let G be a finite group, and let p be a prime. Then G has a Sylow p-subgroup.
Theorem 2.6 (Sylow C). If S and T are Sylow p-subgroups of a finite group G, then
T = Sg for some element g ∈ G.
Theorem 2.7 (Sylow D). Let P be a p-subgroup of a finite group G. Then P is contained in some Sylow p-subgroup of G.
One of the reasons that the Sylow E theorem is considered so important is that it is essentially a partial converse to Lagrange’s theorem. Note that the Sylow
C, D, and E theorems are true in any group G. However, the following theorems, 7 called the Hall E, Hall C, and Hall D theorems (which are similar to the Sylow E,
Sylow C, and Sylow D theorems) require the hypothesis that the group G is solvable.
Theorem 2.8 (Hall E). Suppose that G is a finite solvable group, and let π be an arbitrary set of primes. Then G has a Hall π-subgroup.
Theorem 2.9 (Hall C). Suppose that G is a finite solvable group, and let π be an arbitrary set of primes. Then all Hall π-subgroups of G are conjugate.
Theorem 2.10 (Hall D). Suppose that G is a finite solvable group, and let π be an arbitrary set of primes. Also, let U ⊆ G be a π-subgroup. Then U is contained in some Hall π-subgroup of G.
Returning to a discussion of Sylow p-subgroups, it should be noted that other important subgroups, such as Op(G), are defined in terms of Sylow p-subgroups.
Similarly, if π is a set of primes, Oπ(G) and Oπ′ (G) are defined in terms of Hall
π-subgroups and Hall π′-subgroups, respectively. The formal definitions of these subgroups follow.
Definition 2.11. Suppose G is a group and p is a prime. The subgroup Op(G) is defined as the intersection of all the Sylow p-subgroups of G. Also, the subgroup
′ Op′ (G) is defined as the unique largest normal p -subgroup of G, which means that
′ Op′ (G) contains every normal p -subgroup of G. Similarly, if G is solvable and π is a set of primes, then the subgroup Oπ(G) is defined as the intersection of all the Hall
π-subgroups of G. Also, the subgroup Oπ′ (G) is defined as the intersection of all the
Hall π′-subgroups of G. 8 Some of the subgroups defined above in Definition 2.11 have other equivalent forms. For instance, the subgroup Op(G) can also be defined as the unique largest normal p-subgroup of G, which means that Op(G) contains every normal p-subgroup of G. Similarly, the subgroup Oπ(G) can be equivalently defined as the unique largest normal π-subgroup of G, which means that Oπ(G) contains every normal π-subgroup of G. Finally, the subgroup Oπ′ (G) can also be defined as the unique largest normal
′ ′ π -subgroup of G, which means that Oπ′ (G) contains every normal π -subgroup of G.
The following theorem, which concerns the subgroup Oπ(G) and is known as the Hall-Higman 1.2.3 lemma ([7]), is stated with a slightly stronger hypothesis in order to eliminate some technicalities. This theorem will be used in the proofs of our results.
Theorem 2.12 (Hall-Higman 1.2.3). Let G be a solvable group, and assume that
Oπ′ (G) = 1. Then Oπ(G) ⊇ CG(Oπ(G)).
Another important type of subgroup which can be defined in terms of a Sylow p-subgroup is called a p-complement.
Definition 2.13. Let G be a group and p be a prime. We say a subgroup H of G is a p-complement if H has p-power index and has order not divisible by p. That is,
H is a subgroup of G whose index in G is the order of a Sylow p-subgroup.
It should be noted that p-complements need not exist in an arbitrary group
G.
9 2.2 Solvable Groups and Nilpotent Groups
The following theorem is an important result about solvable groups (recall Definition
1.3) that is used in the initial steps of the group-theoretic proof of Burnside’s theorem.
Theorem 2.14. Let G be any group and suppose N ⊳ G. Then G is solvable if and only if N and G/N are both solvable.
The next definition concerns a class of groups related to solvable groups, called nilpotent groups.
Definition 2.15. A group G is nilpotent if there exists a finite collection of normal subgroups Ni of G such that 1= N0 ⊆ N1 ⊆···⊆ Nr = G and Ni/Ni−1 ⊆ Z(G/Ni−1) for 1 ≤ i ≤ r.
(It should be noted that the Theorem 2.14 does not apply to nilpotent groups.
In other words, if G is any group and N is a normal subgroup of G, then it is not true that if N and G/N are nilpotent, then G is nilpotent.)
The theorem that follows concerns various characterizations of nilpotence which are less technical than the definition and are thus more readily used in a proof than the definition of nilpotence.
Theorem 2.16. Let G be a finite group. The following are equivalent: (i.) G is a nilpotent group, (ii.) NG(H) > H for all H < G, (iii.) every maximal subgroup of
G is normal, (iv.) every Sylow subgroup of G is normal, and (v.) G is the internal direct product of its nontrivial Sylow subgroups.
10 Part (ii.) of the above theorem is often referred to as the property that
“normalizers grow” in nilpotent groups. It should also be noted that parts (ii.), (iv.), and (v.) will be used rather heavily in the proofs of our results.
2.3 Special Types of Subgroups and Elements
The next definition concerns a special type of subgroup, called a p-local subgroup.
Definition 2.17. Let G be a group and p be a prime. Then a subgroup H of G is called p-local if H is of the form H = NG(P ) where P is some nontrivial p-subgroup of G.
The following definition concerns a special type of subgroup, called a char- acteristic subgroup.
Definition 2.18. Let G be a group and let H be a subgroup of G. If H is fixed by every automorphism of G, then we say that H is characteristic in G.
From the definition above, it is easy to see that every characteristic sub- group of G is also normal in G. In addition, we also have the following result about characteristic subgroups.
Theorem 2.19. Let G be a group and let N be a normal subgroup of G. If H is a characteristic subgroup of N, then H ⊳ G.
Our final two definitions concern two special types of elements, called p- central and p′-central elements. The group-theoretic proof of Burnside’s theorem,
11 which will be outlined in the following chapter, heavily utilizes the analysis of p- central elements.
Definition 2.20. Let G be a group and p be a prime. We say an element 1 =6 x ∈ G is p-central if x ∈ Z(P ) for some Sylow p-subgroup P of G.
It is important to notice that p-central elements always exist in any group G where p | |G|, since Sylow p-subgroups are nontrivial p-groups when p | |G| and thus have nontrivial centers, by Theorem 2.3 above.
Definition 2.21. Let G be a group and p be a prime. We say an element 1 =6 x ∈ G is p′-central if x ∈ Z(H) for some p-complement H ⊆ G.
Note that a p-complement need not exist, and if it does, it need not have a nontrivial center. Thus, p′-central elements need not exist.
2.4 Other Necessary Theorems
The final two theorems will be needed in the following chapter for the outline of the group-theoretic proof of Burnside’s theorem.
Theorem 2.22 (Cauchy). Let G be a group and let p be a prime where p | |G|. Then there is an element g ∈ G such that g has order p.
Theorem 2.23. Let t be an involution in a finite group G, where t∈ / O2(G). Then there exists an element x ∈ G of odd prime order such that xt = x−1.
12 CHAPTER III
OUTLINE OF THE GROUP-THEORETIC PROOF OF BURNSIDE’S
THEOREM
Since the proofs of our results invoke the group-theoretic methods used to prove
Burnside’s theorem, it is useful to first understand the structure of the group-theoretic proof due to Goldschmidt [3], Bender [4], and Matsuyama [5] before we discuss the proofs of our results. The following outline concerns the version of the proof found in
[1]. Since certain steps of the proof will be discussed in more detail than others, we encourage curious readers to refer to the full version of the proof for further discussion.
3.1 Preliminary Steps
The proof of Burnside’s theorem is by contradiction and so we consider a group G that is a counterexample of smallest possible order. (Clearly, by assumption, |G| = paqb where p and q are distinct primes.) From this restriction on G, we know that for every subgroup H < G, the order of H is divisible by at most two primes. Thus, by the choice of G as the smallest counterexample, we know that all of these proper subgroups H < G must be solvable (since p-groups are always solvable). Furthermore, if 1 < N ⊳ G, then G/N is solvable as well. Now if N is a nontrivial and proper normal subgroup of G, then by the choice of G we know that both N and G/N are
13 solvable, which implies that G is solvable (by Theorem 2.14) and contradicts that G is a counterexample. Therefore, G cannot have any nontrivial proper normal subgroups and thus G must be simple, by definition. It is important to note that since G is simple, and since every proper subgroup H < G is solvable, G is a minimally simple group, by definition.
Once we have established that the minimal counterexample group G must be simple, we are able to turn our attention to the maximal subgroups of G. Let M be a maximal subgroup of G. Now if 1 Since 1 < K ⊆ M < G, it follows that 1 < K < G and thus K cannot be normal in G since G is simple. Therefore, NG(K) < G and so, by the maximality of M, we have that M = NG(K). Now, since M is maximal, then M is solvable since G is minimally simple. Since M must also be nontrivial, it must have a nontrivial normal subgroup of prime- power order. This implies that either Op(M) > 1 or Oq(M) > 1. Thus, our first major goal is to show that Op(M) and Oq(M) cannot both be nontrivial. 3.2 Analysis of Maximal Subgroups: Steps 1-3 In order to help us accomplish our first major goal, in Step 1 of the proof, we consider a subgroup K ⊆ G that is nilpotent and whose order is divisible by both p and q. We also continue to assume that M = NG(K) where M is a maximal subgroup of G. Our aim is to prove the claim that M is the unique maximal subgroup of G containing K. 14 We begin the proof of Step 1 by assuming to the contrary that K is a maximal counterexample to the claim, so that K ⊆ X and X =6 M is a maximal subgroup of G. Since K is nilpotent, we may write K = Kp × Kq, where Kp and Kq are the nontrivial normal Sylow p-subgroup and Sylow q-subgroup of K, respectively. Since Kp is characteristic in K and K ⊳ M, it follows that Kp ⊳ M and thus M = NG(Kp) since G is simple and M is maximal. Through further analysis of the structure of Kp and Kq, we are able to show that Kp ⊆ Lp and Kq ⊆ Lq, where 1 < Lp = Op(X) and 1 < Lq = Oq(X). By writing L = LpLq, we know that K ⊆ L, where L is a nilpotent group whose order is divisible by both p and q. Also, X = NG(L) is a maximal subgroup of G. Now, in the case where K M = X, which contradicts the choice of X. In the case where K = L, we know that M = NG(K) = NG(L) = X, which again shows that M = X, contradicting the choice of X once more. Therefore, we are able to conclude that M is the unique maximal subgroup of G that contains K, as desired. (It should be noted that our proof of Theorem 4.3 in Chapter 4 is analogus to the above proof of Step 1.) Step 2 focuses on a subgroup P ∈ Sylp(G) that normalizes some nontrivial subgroup V of G. The proof of Step 2 (which is discussed in much greater detail in the proof of Theorem 4.4 in Chapter 4) shows that V cannot be a q-group and that for every Sylow q-subgroup Q of G, we have that hV, Qi = G. Our first major goal is accomplished in Step 3, which shows that if M is maximal in G, then either Op(M) = 1or Oq(M) = 1. This shows that there are 15 exactly two types of maximal subgroups in G, which we call p-type maximal subgroups if Op(M) > 1 or q-type maximal subgroups if Oq(M) > 1. (The proof of Step 3 is mimicked in our proof of Theorem 1.10, found in Chapter 4.) 3.3 Central Elements: Steps 4-6 In Step 4, we begin our analysis of the p-central and q-central elements of G. We suppose that y ∈ NG(V ), where y is q-central and V is a p-subgroup. The proof, which is analogous to the proof of Theorem 4.5 in Chapter 4, shows that V contains no p-central elements. We continue to consider the p- and q-central elements of G in Step 5, whose proof is analogous to the proof of our Theorem 4.6, found in Chapter 4. In this step, our aim is to show that every p-subgroup of G is centralized by a p-central element and that p-type maximal subgroups of G contain no q-central elements. We begin the proof by analyzing a p-subgroup V ⊆ G. By the Sylow E and Sylow D theorems, we are able to choose P ∈ Sylp(G) such that V ⊆ P . It follows that Z(P ) ⊆ CG(V ), which implies that CG(V ) contains a p-central element, by definition. Thus, every p-subgroup of G is centralized by a p-central element, as desired. To finish the proof of the claim in Step 5, we now suppose that M is a p-type maximal subgroup and let V = Op(M). By the conclusion of Step 3 and by the definition of p-type maximal subgroups, we know that V > 1 and Oq(M) = 1. Now, by the Hall-Higman lemma 1.2.3, V ⊇ CM (V )= CG(V ). Thus, V contains a p-central 16 element, since CG(V ) contains a p-central element. Thus, by Step 4, NG(V ) = M contains no q-central elements, as desired. Finally, in Step 6, through repeated applications of the result in Step 5, we are able to show that a q-central element of G cannot normalize a nontrivial p-subgroup, thus completing our analysis of p-central and q-central elements of G. 3.4 G Has Odd Order: Step 7 After our analysis of the p-central and q-central elements of G, we begin Step 7. The goal of this step is to show that p =6 2 and q =6 2. We begin the proof by supposing to the contrary that q = 2 and choosing an involution t, where t is in the center of a Sylow q-subgroup Q of G. (We know that an element t =6 1 exists in the center of a Sylow q-subgroup because Q is a nontrivial q-group and therefore has a nontrivial center, since q is prime. Further, by Cauchy’s Theorem, we know that some involution t must be in the center of Q since q = 2 divides the order of the center.) Since G is simple, and therefore has no nontrivial proper normal subgroups, O2(G) must be trivial and therefore t∈ / O2(G). Thus, by Theorem 2.23 from Chapter 2, it follows that there exists an element x ∈ G of order p such that xt = x−1. It follows that t ∈ NG(hxi). However, since t is q-central, this contradicts the conclusion of Step 6. Therefore, q =6 2. By a similar argument, we can show that p =6 2 as well. 17 3.5 We Obtain a Contradiction: Steps 8-9 Since the conclusion of Step 7 is that p =6 2 and q =6 2, after Step 7 we are able to narrow our focus to groups G with odd order. Thus, at this point, we would be able to appeal to the Feit-Thompson theorem [8], which states that every finite group of odd order is solvable, if we so desired. However, the Feit-Thompson theorem is a very powerful result with an extremely difficult proof, and so we choose to use simpler techniques to finish the proof. In Step 8 of the proof, we consider a p-type maximal subgroup M of G with S ∈ Sylp(M) and show that J(S) ⊳ M and S is a full Sylow p-subgroup of G by using the normal-J theorem. Thus, using our previous notation and the terminology of b(M) and t(M) defined in Chapter 1, Step 8 shows that p ∤ |G : M| and so b(M) ∩ t(M)= ∅. Finally, we obtain a contradiction in Step 9, finishing the proof of Burnside’s theorem, and also showing that there does not exist a minimally simple group G of order paqb. 18 CHAPTER IV RESULTS 4.1 Preliminary Results and Statements of Our Main Results Both of our major theorems about minimally simple groups involve the sets b(M) and t(M) (which were defined in Chapter 1) where M is a maximal subgroup of G. As a preliminary result, notice that we have the following theorem. Theorem 4.1. Suppose G is minimally simple and p-good for all primes p | |G|. Also, let M be a maximal subgroup of G. If p | |G| but p ∤ |M|, then M is nilpotent, and (|M|, |G : M|) = 1. Furthermore, b(M) ∩ t(M)= ∅. Proof. M is nilpotent by the p-goodness of G, since p ∤ |M| and thus Op′ (M) = M. Now, since M is nilpotent, M can be written as M = Kp1 × Kp2 ×···× Kpn , where each Kpi is the unique normal Sylow pi-subgroup of M. To get a contradiction, suppose to the contrary that for some pj, we have that ⊳ pj | |M| and pj | |G : M|. Thus, 1 Since pj | |G : M| we can let P ∈ Sylpj (G) be such that Kpj < P . Thus, since P is nilpotent and proper subgroups of nilpotent groups are properly contained in their normalizers, NP (Kpj ) > Kpj . Thus, NP (Kpj ) * M, since Kpj ∈ Sylpj (M). 19 Therefore, NG(Kpj ) * M. This is a contradiction, since M = NG(Kpj ), and thus (|M|, |G : M|) = 1. To show that b(M) ∩ t(M)= ∅, notice that since M is nilpotent, Oq(M) > 1, for all primes q such that q | |M|. Thus, b(M) consists of all the primes q such that q | |M|, by definition. However, t(M) = {q prime: q | |G : M|} by definition. Therefore, since (|M|, |G : M|) = 1, there can be no primes in common between b(M) and t(M), thus showing that b(M) ∩ t(M)= ∅. We will later prove the following related result (previously stated in Chapter 1), which removes the p-goodness restriction. Theorem 1.9. If G is minimally simple and M is a maximal subgroup of G, with |G : M| = qn, where q is a prime, then b(M) ∩ t(M)= ∅. We therefore have the following related conjecture: Conjecture 1.8. Let G be a minimally simple group. Then for every maximal sub- group M of G, we have |b(M) ∩ t(M)| ≤ 1. Notice that we do not suggest that b(M) ∩ t(M)= ∅ in the conjecture above, since there is an example where |b(M) ∩ t(M)| = 1 (see Chapter 5 for evidence of such an example). In the search of a proof or counterexample for the above conjecture, we have proven the following two theorems, which are our main results for minimally simple groups. 20 Theorem 1.10. Let G be a minimally simple group that is p-good for all primes p dividing |G|, and let M be a maximal subgroup of G. Then |b(M) ∩ t(M)| ≤ 1. Theorem 4.2 (Revised Theorem 1.11). Let G be a minimally simple group that is p-good for all primes p dividing |G|, and let M be a maximal subgroup of G. Suppose b(M) ∩ t(M) = {p}. Let π = b(M) \ {p}. Then G has a Hall π-subgroup contained in M. Moreover, if π is empty, then M contains no p′-central elements. Recall that p′-central elements were defined in Chapter 2 as elements 1 =6 x ∈ G where x ∈ Z(H) for some p-complement H ⊆ G. Also, it should be noted that Theorem 4.2 above is a stronger, but more technical, result than the version of Theorem 1.11 that was stated in Chapter 1. In order to prove the above two theorems, however, we will need to prove some additional theorems, which follow. This first theorem will be used in the proof of Theorem 1.10 and is analogous to Step 1 in the proof of Burnside’s Theorem found in [1]. Theorem 4.3. Suppose G is minimally simple and p-good for each prime p such that p | |G|. Assume K ⊆ G is nilpotent and that M = NG(K) is a maximal subgroup of G. If |K| is divisible by more than one prime, then M is the unique maximal subgroup of G containing K. Proof. This argument mimics the proof in [1] of Step 1 in the proof of Burnside’s Theorem. 21 Suppose not, that is, suppose that there is another maximal subgroup of G containing K that is not M. Let K be the largest counterexample subgroup. So, let K ⊆ X, where X is maximal in G and X =6 M. Notice that since K is nilpotent, if p is a prime, then K can be written as K = Kp × Kp′ , where Kp is the normal Sylow p-subgroup of K and Kp′ is a normal p-complement of K. Claim. If p is a prime such that p | |K| and we write K = Kp × Kp′ , then Kp ⊆ Op(X). Proof of Claim. Let p be any prime such that p | |K| and let q be a prime such that q =6 p and q | |K|. (By assumption, there is such a prime q.) Since K is nilpotent, we can write K = Kq × Kq′ . Notice that Kq is characteristic in K and K ⊳ M, and thus Kq ⊳ M. Thus, due to the maximality of M and the fact that G is simple, M = NG(Kq). Also, M ∩ X = NX (Kq) is a q-local subgroup of X, by definition. By a similar argument, notice that Kp ⊳M = NG(Kp) and M ∩X = NX (Kp). ′ Thus, Kp ⊳M ∩X. In particular, Kp ⊆ Oq′ (M ∩X), since Kp is a normal q -subgroup of M ∩ X. Now since G is minimally simple, X is solvable, and thus X is also q-solvable, and recall that we showed M ∩ X is a q-local subgroup of X. Thus, we can apply Theorem 4.33 of [1] to see that Oq′ (M ∩ X) ⊆ Oq′ (X). Since Kp ⊆ Oq′ (M ∩ X), we clearly now also have Kp ⊆ Oq′ (X). 22 By assumption, since X is maximal in G and G is q-good, Oq′ (X) is nilpotent. Thus, we may write Oq′ (X)= Lp × Lp′ , where Lp is the normal Sylow p-subgroup of ′ Oq′ (X) and Lp′ is a normal p -complement of Oq′ (X). ′ We now argue that Lp = Op(X). This is true since Op(X) is a normal q - subgroup of X and is a p-group. Thus, Op(X) ⊆ Oq′ (X) and Op(X) is contained in Lp, by definition of Lp. Now since Lp is a normal p-subgroup of X, by definition of Op(X) as the largest normal p-group of X, Lp must also be contained in Op(X). Therefore, Lp = Op(X), as claimed. Thus, Kp ⊆ Oq′ (X)= Lp × Lp′ = Op(X) × Lp′ , and so Kp ⊆ Op(X) since Kp is a p-group. The claim is now proven. To resume the proof of the theorem, let L = Lq1 × Lq2 ×···× Lqn , where the qi are the primes that divide |K| and Lqi = Oqi (X). Since Kqi ⊆ Lqi for all i by the claim, K = Kq1 × Kq2 ×···× Kqn ⊆ L. Thus, we have that L is nilpotent (since it is the direct product of its normal Sylow qi-subgroups, by construction) and ⊳ L X as well, since L is the direct product of Oqi (X) terms which are normal in X, by definition. Also, |L| is divisible by at least two primes and K ⊆ L. Case I: Suppose K = L. Then M = NG(K)= NG(L)= X, since L ⊳ X and X is maximal and G is simple. This is a contradiction, however, since M =6 X. Case II: Suppose K 23 Let p and q be distinct primes dividing |K|. Now, Lp ⊆ CG(Lq) ⊆ CG(Kq), since Kq ⊆ Lq by the claim. Furthermore, CG(Kq) ⊆ NG(Kq) = M. Thus, Lp ⊆ M for all primes p such that p | |K|, and so L = Lq1 × Lq2 ×···× Lqn ⊆ M, by construction of L. Thus, L ⊆ M. However, M was chosen such that M =6 X. Since M is maximal in G, this contradicts the assumption that X is the unique maximal subgroup of G containing L. Therefore, in both cases, we have a contradiction. 4.2 Results That Do Not Require p-Goodness All of the results in this section do not require the p-goodness assumption. Our next result is analogous to Step 2 in the proof of Burnside’s Theorem found in [1]. Theorem 4.4. Suppose G is simple and for a given prime q, G has a q-complement P . Suppose P normalizes some nontrivial subgroup V of G. Then hV, Qi = G for every Sylow q-subgroup Q of G. In particular, V cannot be a q-group. Proof. The argument that follows is the same argument with the same notation as provided by Isaacs [1] in the proof of Step 2 of the proof of Burnside’s Theorem. We have provided some additional details on the reader’s behalf. Let Q ∈ Sylq(G), and write H = hV, Qi. Now, since P is a q-complement of G, and since P ∩ Q = 1, |P Q| = |P ||Q| = |G|, and thus P Q = G. 24 Now, let g be an arbitrary element of G. By the above, we can write g = xy, where x ∈ P and y ∈ Q. Since P normalizes V , we have V g = V xy = V y ⊆ H, and thus H contains all G-conjugates of V . Thus, since the nonidentity subgroup V G is generated by the conjugates of V , V G ⊆ H. Also notice that V G ⊳ G. Now, since G is simple, and 1 =6 V G ⊳G, we see that V G = G, and so H = G, since G = V G ⊆ H. Thus, H = hV, Qi = G for every Sylow q-subgroup Q of G, as claimed. Finally, to show that V cannot be a q-group, notice that if V is a q-group, then we can choose Q to contain V , by the Sylow-D Theorem. Then hV, Qi = Q < G, which is a contradiction to the above. The following result will be used in the proof of Theorem 4.7 and is analogous to Step 4 in the proof of Burnside’s Theorem found in [1]. Theorem 4.5. Suppose G is simple and q is a prime such that G contains a q- ′ complement. Suppose y ∈ NG(V ), where y is q -central and V is a q-subgroup. Then V contains no q-central elements. Proof. The argument that follows is the same argument with the same notation as provided by Isaacs [1] in the proof of Step 4 of the proof of Burnside’s Theorem. We have provided some additional details on the reader’s behalf. Given a q-subgroup U ⊆ G, write U ∗ to denote the subgroup generated by the q-central elements in U. Since the G-conjugates of a q-central element are q- 25 central, it follows that NG(U) permutes the q-central elements of U, and thus NG(U) normalizes U ∗. ∗ In particular, since y ∈ NG(V ), y also normalizes V , and our goal is to show V ∗ = 1. Now V ∗ is a q-subgroup of G that is normalized by y and generated by q- central elements. We let W be a q-subgroup of G, maximal subject to the conditions that W is normalized by y and generated by q-central elements. If V ∗ > 1, then W > 1, and we work to obtain a contradiction. ′ Write N = NG(W ), and let S ∈ Sylq(N) so that S ⊇ W . Since y is q -central, we can choose Q, a q-complement such that y ∈ Z(Q). Then hyi is normalized by Q, and since y ∈ N, we have hy,Si ⊆ N < G, where the strict inequality holds because 1 It follows by Theorem 4.4 that if S ∈ Sylq(G) we have a contradiction. Thus, S is not a full Sylow q-subgroup of G. Since S ∈ Sylq(N) and S is not a full Sylow q-subgroup of G, S Now, since W is generated by q-central elements and W g =6 W , there must be at least one q-central generator x of W such that xg ∈/ W . Because x is q-central, 26 we can choose R ∈ Sylq(G) such that x ∈ Z(R). Now, G = RQ, and so we can write g = ab, with a ∈ R and b ∈ Q. Then xg = xab = xb ∈ W b and xg ∈ W g ⊆ N, and thus xg ∈ W b ∩ N. Since W b ∩ N is a q-subgroup that normalizes the q-group W , it follows that W (W b ∩ N) is a q-subgroup of G. Also, W = W ∗, and thus (W (W b ∩ N))∗ ⊇ hW, xgi >W . Now W b is normalized by yb = y. Since y ∈ N, it follows that W b ∩ N is normalized by y, and therefore (W (W b ∩ N))∗ is a q-subgroup that is normalized by y. This subgroup is generated by q-central elements and it strictly contains W . This is a contradiction since it contradicts the choice of W . Our next result is the starting point for analyzing the sets b(M) and t(M) of maximal subgroups M of minimally simple groups G. Notice that we do not make any assumption of p-goodness. Theorem 4.6. Suppose that G is minimally simple and that M is a maximal subgroup n of G, with |G : M| = q , where q is a prime. Then Oq(M) = 1. Proof. Suppose not. That is, suppose that Oq(M) =6 1. Now, since M is solvable, M contains a Hall q′-subgroup H, by the Hall-E Theorem. So, |M : H| = ql for some nonnegative l ∈ Z, and |H| is a q′-number. Thus, it is clear that H is a q-complement in G. Notice that H ⊆ NM (Oq(M)), since Oq(M) ⊳ M, by definition. Now, if we let H = P in the statement of Theorem 4.4, we can also let V = Oq(M) since V =6 1, 27 by assumption. However, this contradicts Theorem 4.4, since V is a q-group, by definition. Thus, V = Oq(M) = 1, as claimed. The proof of Theorem 1.9, which uses the above result and specifically men- tions the sets b(M) and t(M), now follows. Theorem 1.9. If G is minimally simple and M is a maximal subgroup of G, with |G : M| = qn, where q is a prime, then b(M) ∩ t(M)= ∅. Proof. Since |G : M| = qn, t(M) = {q}, by definition. However, by Theorem 4.6, under these conditions, Oq(M) = 1. Thus, by definition, q∈ / b(M). Therefore, b(M) ∩ t(M)= ∅. The result that follows will be used in the proof of Theorem 4.2 and is anal- ogous to Step 5 in the proof of Burnside’s Theorem found in [1]. Theorem 4.7. Suppose G is a minimally simple group, M is a maximal subgroup of ′ G, and that Oq′ (M) = 1, where q is a prime. Then M contains no q -central elements of G. Proof. This argument mimics the proof in [1] of Step 5 in the proof of Burnside’s Theorem. Given a q-subgroup V ⊆ G, choose Q ∈ Sylq(G) with Q ⊇ V . Then 1 < Z(Q) ⊆ CG(V ), and so CG(V ) contains a q-central element, by definition. Now suppose that M is maximal in G and Oq′ (M) = 1 and let V = Oq(M). V is nontrivial since M is solvable and G is minimally simple. 28 Thus, by the Hall-Higman 1.2.3 theorem, V = Oq(M) ⊇ CM (Oq(M)) = CG(Oq(M)) = CG(V ). Here, the second equality is a consequence of the following: If we let x ∈ CG(Oq(M)), then x ∈ NG(Oq(M)) = M since G is simple and M is maximal in G. Thus, since x ∈ M as well, x ∈ CM (Oq(M)). So, CM (Oq(M)) ⊇ CG(Oq(M)). The other containment is clear, by definition. Now, since V ⊇ CG(V ), V contains a q-central element since CG(V ) contains a q-central element. If we now suppose that G does not contain a q′-central element, then the result is trivial, since there cannot be a q′-central element in M. So, suppose that G contains a q′-central element. Thus, G contains a q-complement, by definition. Now, to obtain a contradiction, suppose that M contains a q′-central element y. Since V ⊳ M, y ∈ NG(V ). However, by Theorem 4.5, V cannot have q-central elements, which we have shown is not the case. This is a contradiction. 4.3 Final Main Results We are now prepared to prove the following major result about minimally simple groups. It should be noted that the results in this section require the p-goodness assumption. Theorem 1.10. Let G be a minimally simple group that is p-good for all primes p dividing |G|, and let M be a maximal subgroup of G. Then |b(M) ∩ t(M)| ≤ 1. 29 Proof. This argument mimics the proof in [1] of Step 3 in the proof of Burnside’s Theorem. Suppose that |b(M) ∩ t(M)| > 1. That is, suppose that there are distinct primes p,q ∈ b(M) ∩ t(M) and write Zp = Z(Op(M)) and Zq = Z(Oq(M)). Notice that, by definition of b(M), both Zp and Zq are the centers of nontrivial p- and q- groups, respectively. Thus, both Zp and Zq are nontrivial and characteristic in M. Also, Zp and Zq are abelian by construction and centralize each other since they intersect trivially. Therefore, the group Z = ZpZq is abelian. Moreover, Z ⊳ M, and so M = NG(Z), since M is maximal in G and G is simple. Now, since Z is abelian, Z is nilpotent, and also, by construction, |Z| is divisible by more than one prime. Thus, by Theorem 4.3, M is the unique maximal subgroup of G containing Z. Now if 1 =6 z ∈ Z, then Z ⊆ CG(z) < G, since G is simple, and so CG(z) is contained in some maximal subgroup of G that contains Z, and thus CG(z) ⊆ M. Let S ∈ Sylp(M). By the definition of t(M) and since p ∈ b(M) ∩ t(M), S is not a full Sylow p-subgroup of G. Thus, S < P for some Sylow p-subgroup P of G, and hence NP (S) is a p-subgroup of G strictly larger than S, since P is nilpotent and proper subgroups of nilpotent groups are properly contained in their normalizers. Now, since S ∈ Sylp(M), and NP (S) > S is a p-subgroup, we have NP (S) * M. Thus, we can choose an element g ∈ G − M such that Sg = S. 30 The argument that follows from here is the same argument with the same notation as provided by Isaacs [1]. We have provided some additional details on the reader’s behalf. g Now write A = (Zp) and notice that since Zp ⊆ Op(M) ⊆ S, we have g g A =(Zp) ⊆ S = S ⊆ M, and thus A acts on the normal subgroup Zq of M. We now show that the action of A on Zq is faithful and that A is cyclic by showing that we get a contradiction if we assume otherwise. Suppose either that the action of A on Zq is not faithful or that A is not cyclic. In both cases, we claim that Zq = hCZq (a) | 1 =6 a ∈ Ai. This is clear if the action is not faithful, since in that case there exists a nonidentity element a ∈ A such that CZq (a) = Zq. If A is not cyclic, then since A is abelian, Theorem 6.21 in [1] applies to yield the claim. g g Now since CG(z) ⊆ M for all z ∈ Z such that z =6 1, CG(z ) ⊆ M for all g g g g g g z ∈ Z such that z =6 1. In particular, since A =(Zp) ⊆ Z , we have CG(a) ⊆ M if 1 =6 a ∈ A. If the action of A on Zq is not faithful or if A is not cyclic, therefore, we g g deduce that Zq ⊆ M since Zq = hCZq (a) | 1 =6 a ∈ Ai⊆hCG(a) | 1 =6 a ∈ Ai ⊆ M . g g g Also, Zp ⊆ S = S ⊆ M , and thus, Z = ZpZq ⊆ M . But M is the unique maximal g subgroup containing Z, and therefore M = M and we have g ∈ NG(M)= M (since G is simple), which is not the case, since g ∈ G − M. This contradiction shows that the action of A on Zq is faithful and that A is cyclic. Thus, we may assume that A is cyclic and acts faithfully on Zq. Also, since g A = (Zp) , we see that Zp is cyclic. Now, by similar reasoning, with the roles of p and q interchanged, we deduce that Zq is cyclic also. 31 However, A is a nontrivial p-group that acts faithfully on Zq, and so p divides |Aut(Zq)|, and since Zq is cyclic, we see that p | (q − 1) and so p Our next and final theorem is a result of studying the case when |b(M) ∩ t(M)| = 1. Theorem 4.2 (Revised Theorem 1.11). Let G be a minimally simple group that is p-good for all primes p dividing |G|, and let M be a maximal subgroup of G. Suppose b(M) ∩ t(M) = {p}. Let π = b(M) \ {p}. Then G has a Hall π-subgroup contained in M. If π is empty, then M contains no p′-central elements. Proof. Suppose π is nonempty, so that Op′ (M) > 1. By p-goodness, Op′ (M) is nilpotent. Let q1,q2,...qm be the primes that divide |Op′ (M)|. Thus, by nilpotence, ′ we may write Op (M)= Kq1 × Kq2 ×···× Kqm , where Kqi is the (nontrivial) normal Sylow qi-subgroup of Op′ (M). Claim. Kqi = Oqi (M). Proof of Claim. Let qi be any prime such that the normal Sylow qi-subgroup ′ ′ ′ of Op (M) is nontrivial. By the nilpotence of Op (M), we have Op (M) = Kqi × ′ ′ ′ Kqi , where Kqi is the normal Sylow qi-subgroup of Op (M) and Kqi is a normal qi- complement of Op′ (M). ′ Now, Oqi (M) is a normal p -subgroup of M and is a qi-group. Thus, Oqi (M) ⊆ ′ Op (M) and Oqi (M) ⊆ Kqi , by definition of Kqi . 32 Also, we have that Kqi ⊆ Oqi (M), since Kqi is a normal qi-subgroup of M and Oqi (M) is defined as the unique largest normal qi-subgroup of M. Therefore, Kqi = Oqi (M), as claimed, and the claim is proven. To resume the proof of the theorem, notice that the claim shows that q1,q2,...,qm ∈ b(M). Also, π = {q1,q2,...,qm}. Now, since b(M) ∩ t(M) = {p} by assumption, q1,q2,...,qm are not top primes. Also, since M is solvable, then M has a Hall π-subgroup, by the Hall-E Theorem. By definition of t(M), we have q1,q2,...,qm ∤ |G : M|. Due to this and the fact that M has a Hall π-subgroup, G must have a Hall π-subgroup contained in M. Suppose now that π = ∅. This is equivalent to Op′ (M) = 1. In this case, by Theorem 4.7, M contains no p′-central elements. 33 CHAPTER V GAP RESULTS 5.1 Method Recall that Thompson classified all the minimally simple groups into five major “types” (up to isomorphism) in his paper [6]. Thompson’s five types of minimally simple groups are: (i.) PSL(2, 2p), where p is any prime, (ii.) PSL(2, 3p), where p is any odd prime, (iii.) PSL(2,p), where p> 3 is any prime where p2 + 1 ≡ 0 (mod 5), (iv.) Sz(2p), where p is any odd prime, and (v.) PSL(3, 3). Thus, it is possible to use the computer program GAP to analyze some of the minimally simple groups in order to search for evidence of the strength of our major results. As we mentioned in Chaper 1, we were able to use GAP to show that Theorem 1.10 is the strongest possible result of its type, because we have found an example where |b(M)∩t(M)| = 1 for some maximal subgroup M of a minimally simple group G. We were able to do this by finding b(Mi) and t(Mi) for all the maximal subgroup class representatives Mi for the minimally simple groups we analyzed. 34 5.2 Description of Tables Tables 5.1, 5.2, and 5.3 below display our results from the analysis of some mini- mally simple groups using GAP. The first column displays the name of the minimally simple group (call it G for the purposes of these tables and this discussion) and the second column contains the names of all the maximal subgroup class representatives (notated as Mi, i = 1, 2, 3,... ) of G, which were found using the “MaximalSubgroup- ClassReps(G)” command in GAP. We begin the analysis of b(Mi) and t(Mi) by first finding the order of each subgroup Mi, which is displayed in the third column, and computing b(Mi) and t(Mi) from their definitions (recall Definition 1.7) and display- ing the results in columns four and five, respectively. Finally, we display the set b(Mi) ∩ t(Mi) for each Mi in the sixth column. From the tables, it is clear that (with the three exceptions of PSL(2, 32), PSL(2, 128), and PSL(2, 13)1) most of the minimally simple groups G we analyzed are p-good for all primes p | |G|. (This is true because the order of each Mi is divisible by two or fewer primes, which means that, for all primes p | |G|, the subgroup Op′ (Mi) is a q-group, where q =6 p is a prime, and hence, Op′ (Mi) is nilpotent. Thus p-goodness follows by definition.) However, by using GAP, we have shown that PSL(2, 13) is indeed not p-good for some prime p | |PSL(2, 13)|, which creates a distinction between the hypotheses in Conjecture 1.8 and Theorem 1.10. Also, upon further inspection 1Calculations with GAP can be used to show that P SL(2, 13) is in fact not p-good for some prime p | |P SL(2, 13)|. Also, at this time we are unable to show p-goodness with GAP for P SL(2, 32) and P SL(2, 128) due to the limitations of GAP’s ability to compute. 35 of the tables, it is clear that PSL(2, 17) (as well as PSL(3, 3)) is an example of a minimally simple group where, in particular, |b(M3) ∩ t(M3)| = 1, thus showing that Theorem 1.10 is the strongest possible result of its type. It should be noted that an asterisk in the tables refers to a case where we were unable to use GAP to analyze an aspect of the group due to the limitations of GAP’s ability to compute. 36 5.3 Tables Table 5.1: Groups of the Type PSL(2, 2p) Group Max. Subgroups Mi |Mi| b(Mi) t(Mi) b(Mi) ∩ t(Mi) M1 6 = 2 × 3 {3} {2, 5} ∅ 1 PSL(2, 4) M2 10 = 2 × 5 {5} {2, 3} ∅ 2 M3 12 = 2 × 3 {2} {5} ∅ M1 14 = 2 × 7 {7} {2, 3} ∅ 2 2 PSL(2, 8) M2 18 = 2 × 3 {3} {2, 7} ∅ 3 M3 56 = 2 × 7 {2} {3} ∅ M1 62 = 2 × 31 {31} {2, 3, 11} ∅ 3 PSL(2, 32) M2 66 = 2 × 3 × 11 * {2, 31} * 5 M3 992 = 2 × 31 * {3, 11} ∅ M1 254 = 2 × 127 * {2, 3, 43} * 4 PSL(2, 128) M2 258 = 2 × 3 × 43 * {2, 127} * 7 M3 16, 256 = 2 × 127 * {3, 43} ∅ 37 Table 5.2: Groups of the Type PSL(2, 3p) and PSL(2,p) Group Max. Subgroups Mi |Mi| b(Mi) t(Mi) b(Mi) ∩ t(Mi) 2 M1 12 = 2 × 3 {2} {3, 7, 13} ∅ 5 PSL(2, 27) M2 26 = 2 × 13 {13} {2, 3, 7} ∅ 2 M3 28 = 2 × 7 {2, 7} {3, 13} ∅ 3 M4 351 = 3 × 13 * {2, 7} ∅ M1 21 = 3 × 7 {7} {2, 3} ∅ 3 6 PSL(2, 7) M2 24 = 2 × 3 {2} {7} ∅ 3 M3 24 = 2 × 3 {2} {7} ∅ 2 M1 12 = 2 × 3 {2} {7, 13} ∅ 2 7 PSL(2, 13) M2 12 = 2 × 3 {2, 3} {7, 13} ∅ M3 14 = 2 × 7 {7} {2, 3, 13} ∅ M4 78 = 2 × 3 × 13 {13} {2, 7} ∅ 4 M1 16 = 2 {2} {3, 17} ∅ 2 M2 18 = 2 × 3 {3} {2, 17} ∅ 3 8 PSL(2, 17) M3 24 = 2 × 3 {2} {2, 3, 17} {2} 3 M4 24 = 2 × 3 {2} {2, 3, 17} {2} 3 M5 136 = 2 × 17 {17} {2, 3} ∅ 38 Table 5.3: PSL(2, 23), Sz(8), and PSL(3, 3) Group Max. Subgroups Mi |Mi| b(Mi) t(Mi) b(Mi) ∩ t(Mi) M1 22 = 2 × 11 {11} {2, 3, 23} ∅ M2 253 = 11 × 23 {23} {2, 3} ∅ 3 9 PSL(2, 23) M3 24 = 2 × 3 * {11, 23} ∅ 3 M4 24 = 2 × 3 * {11, 23} ∅ 3 M5 24 = 2 × 3 * {11, 23} ∅ M1 14 = 2 × 7 * {2, 5, 13} * 2 10 Sz(8) M2 20 = 2 × 5 * {2, 7, 13} * 2 M3 52 = 2 × 13 * {2, 5, 7} * 6 M4 448 = 2 × 7 * {5, 13} ∅ 3 M1 24 = 2 × 3 {2} {2, 3, 13} {2} 11 PSL(3, 3) M2 39 = 3 × 13 {13} {2, 3} ∅ 4 3 M3 432 = 2 × 3 * {13} ∅ 4 3 M4 432 = 2 × 3 * {13} ∅ 39 5.4 Possible Future Research Since we have discovered that PSL(2, 13) is a minimally simple group that is not p-good for some prime p | |G|, our major goal for the future concerns developing a proof for Conjecture 1.8, which is essentially a version of Theorem 1.10 without the p-goodness assumption. In particular, we may be able to use methods similar to those found in Burnside’s original character theory proof of his paqb theorem [2] to gain some insight into the validity of Conjecture 1.8. Additionally, inspection of Tables 5.1, 5.2, and 5.3 leads us to the natural conjecture that b(M) ∩ t(M) ⊆ {2}. Thus, in the future, we may also focus on proving this conjecture or finding cases where an odd prime p is an element of the set b(M) ∩ t(M). 40 BIBLIOGRAPHY [1] I. Martin Isaacs. Finite group theory. American Mathematical Society, Providence, Rhode Island, 1st edition, 2008. [2] William Burnside. On groups of order pαqβ. Proceedings of the London Mathe- matical Society, 1:388–392, 1904. [3] David M. Goldschmidt. A group theoretic proof of the paqb theorem for odd primes. Mathematische Zeitschrift, 113:373–375, 1970. [4] Helmut Bender. A group theoretic proof of Burnside’s paqb-theorem. Mathema- tische Zeitschrift, 126:327–338, 1972. [5] Hiroshi Matsuyama. Solvability of groups of order 2apb. Osaka Journal of Math- ematics, 10:375–378, 1973. [6] John G. Thompson. Nonsolvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74:383–437, 1968. [7] P. Hall and Graham Higman. On the p-length of p-soluble groups and reduction theorems for Burnside’s problem. Proceedings of the London Mathematical Society Third Series, 6:1–42, 1956. [8] Walter Feit and John G. Thompson. Solvability of groups of odd order. Pacific Journal of Mathematics, 13:775–1029, 1963. 41