On Local Formations of Finite Groups by Lam Chak Ming A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Mathematics ⑥The Chinese University of Hong Kong September 2001 The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. |( I 3 APR M 1 fejTUNIVERSITY /My/ 'HUBRARY SYSTEMy^/ 摘要 在代數範疇中，群論是一個歷史悠久的分支，而有限可解群在有 限群理論中更佔有顯著的地位。在過去數十年，有限可解群的理論， 在多位數學家的努力下獲得許多重要的成果。例如，E. Zelmanov， 他用上許多可解群的技巧來處理著名的Restricted Burnside Problem，並因此在1994年獲得菲爾茲獎。隨著可解群理論的成 熟，個別的群類如Fitting類、Schunk類，群系等亦得以發展。 在 1963 年，W.Gaschiitz 發表了 一篇文章名為 “Zur Theorie der endlichen aufbsbamn Gruppen”奠定了群系理論的基礎。隨後， H.Neumann, L.A. Shemetkov, A. Skiba等人亦運用了群系的技巧去 解決群論的問題。這對群系理論的發展有相當的貢獻。除此之外， 群系理論亦間接地影響無窮群、李代數的分類等發展。群系理論的方 法及其技巧，最近亦應用到一般代數方面。 在本論文中，我們會先介紹群系的基礎知識；另外，一些基本的 概念如上根，f超中心，^r-覆蓋子群及f正規化子等亦會討論。 在論文的後半部份，我們將討論局部群系及其應用。我們會利用 群系函數去構造群糸。基本的局部群糸的性質亦會討論。我們亦會應 用局部群系的技巧去研究Schmidt群的結構。例如，我們會證明某個 Schmidt群的Frattini子群相等於那個Schmidt群的超中心；以及一 個Schmidt群的N-上根的Frattini子群是那個N-上根的子群。最 後，我們將利用這些曾探討過的技巧來推廣有關P-冪零群的伊籐定 理（lto，s Theorem)。這是一個新的結果。 On Local Formations of Finite Groups 1 Abstract The theory of groups is one of the main branches in algebra with a long history and finite soluble groups form a remarkable subclass of the class of finite groups. In the past decade, many authors obtained significant results of soluble groups. For example, Efim Zelmanov has used some results in soluble groups to solve the famous Restricted Burnside Problem. Along with soluble groups, certain classes of finite groups, like Fitting class, Schunk class and formation, were also developed. The theory of formation was first introduced by "Zur Theorie der endlichen auflosbaren Gruppen" in 1963. Later on, H. Neumann, L.A. Shemekov, A. Skiba and T. Hawkes also adopted the theory of formation to solve some problems in group theory. The theory of formation of finite groups has later been applied to the theory of infinite groups, Lie algebras and universal algebras. In this thesis, we will first give a survey on the theory of formations and intro- duce some basic concepts such as J^-coradicals, jF-hypercentres and ^-"-covering subgroups. Also, we will discuss the ^"-normalizers and prove some useful results that are related to JF-normalizers. In the later part of the thesis, we will focus on the theory of local formations. We will show how to construct a local formation by using a formation function. Some useful properties of local formations will be consequently obtained. In par- ticular, we will use these properties of local formations to describe the structure of Schmidt groups. For example, we will prove that the Frattini subgroup of a Schmidt group is equal to its hypercentre and also the Frattini subgroup of the A/*-coradical of a Schmidt group is contained in the centre of its AA-coradical. Finally, we shall study Ito's theorem with a weaker condition. A generalized version of Ito's theorem of p-nilpotent, groups is obtained. Also, some possible generalized results related to local formation of finite groups are considered. On Local Formations of Finite Groups 2 ACKNOWLEDGMENTS I wish to express my gratitude to Prof. K.P. Shnm for his supervision. His stimulating guidance and encouragement has helped me a lot during the prepara- tion of this thesis. I would also like to thank Prof. Guo W.B. for his helpful advise. Lam Chak Ming The Chinese University of Hong Kong July, 2001 Contents 0 Introduction 5 1 Background Knowledge of Group Theory 7 1.1 Some Basic Results 7 1.2 Solvable Groups 13 1.3 Nilpotent groups and some useful results 15 2 Theory of Formations 21 2.1 Some Basic Results of Formations 21 2.2 ^-covering subgroups and 义-projectors 23 2.3 The Conjugacy of ^"-Covering Subgroups 32 2.4 JF-Normalizers 36 3 Local Formations 47 3.1 The Construction of Local Formations 47 3.2 The Stability of Formations 51 3.3 The Complements of JT-coradicals 57 3.4 Minimal non-JT-groups 59 3 On Local Formations of Finite Groups 4 4 Finite Groups With Given Minimal Subgroups 66 4.1 C-normality of Groups 66 4.2 A Generalized Version of Ito's Theorem 70 Bibliography 76 Chapter O Introduction One of the most important theorem in finite group theory is the theorem of Sylow which states the following: Existance: Every finite group G possesses Sylow p-groiips; Conjugacy: The Sylow p-groups of G form a conjugacy class of G] Dominance: Each p-subgroup of G is contained in a Sylow p-subgroup. For solvable groups, P. Hall was able to extend the scope of Sylow's theorem from p-groups to 7r-groups for a set tt of primes. In 1961, R.W. Carter discovered that every finite solvable group has self-normalizing nilpotent groups (known as Carter subgroups). The discovery of Carter subgroups gave a considerable fillip to the subsequent developments in solvable groups. Besides, the ideas that come from Cater subgroups also invite analogy with Cartan subalgebras, which play a part in the classification of finite dimensional, semisimple Lie algebras. In 1963’ Gaschiitz in his work “ Zur Theorie der endlichen auflosbaren Grup- pen" presented remarkably fruitful extensions of Carter's ideas. In his work, he introduced the idea of saturated formation. Besides, he defines the concept of 5 On Local Formations of Finite Groups 6 an ^"-covering subgroup and he showed that for any staurated formation T, T- covering subgroups exist and form a conjugacy class in each finite solvable group. By specializing T to MV, SIX and M in turn, one obtains the Sylow, Hall, Carter subgroups respectively. Furhtermore, one can show that a rich supply of exam- ples can be obtained from the construction of local formations. These ideas are the source of inspiration for much subsequent work. In this thesis, we will study the theory of formations. Some basic concepts such as 厂coradicals，J^-hypercentres, J^-covering subgroups and their related properties will be introduced. Besides, we will employ the technique of local formations to prove a generalized version of Ito's theorem on p-nilpotent, groups. Some criteria for the solvability of a group will also be discussed. The following is the arrangement of this thesis. In chapter 1, we will cover the properties of solvable groups and nilpotent groups. Some technical lemmas will be discussed. Theory of formations will be discussed in chapter 2. We will first give some basic results and then we will give the definition of ；^-covering subgroups and ^-projectors. The conjugacy of A'-covering subgroups and A'-normalizers will also be discussed. In chapter 3, we will discuss the theory of local formation. The structure of minimal non-J^-groups will also be considered. In particular, we will use some properties of local formations to study the structure of minimal non-A/"- groups(Schmidt groups). In the last chapter, we will use the technique developed in the previous chap- ters to study the structure of some finite groups. First, we will use the properties of a Schmidt group to prove a generalized version of Ito's theorem on p-nilpotent groups. Besides, we will introduce the concept of c-normality of groups. We will use the c-normality of groups to give some criteria for the solvability of groups. In this thesis, all groups are supposed to be finite. Chapter 1 Background Knowledge of Group Theory 1.1 Some Basic Results Definition 1.1.1 A subgroup M of a group G is said to be m,aximal if M ^ G •d if there exists a subgroup T of G satisfying M < T < G, then T = M or T = G. Definition 1.1.2 Let G be a group. The Frattini subgroup of the group G ？:s defined to be the intersection of all the maximal subgroups of G. If G has no maximal subgroup, then we define ^(G) 二 G. Definition 1.1.3 Let G be a group and x G G. Then x is called a non generator of G if G =< T U {.t} > implies that G =<T >, where < T > is the subgroup generated by T Theorem 1.1.4 fFrattini) Let G be a non-trival group. Then the Frattini sub- group of G is the set of all nongenerators of G. Proof: Let x be a non-generator of G. Suppose that x 突 Then there exists a maximal subgroup M such that x 雀 M. Hence, G =< {x}uM >=< M〉二 M1 7 On Local Formations of Finite Groups 12 a contradiction. This leads to .T G 否(G). NOW, suppose that .T is not a non- generator of G. Then there exists a subset, TofG such that < Tu{.r} >二 G and <T >= H ^ G. This leads to H < M for some maximal subgroup M of G. If x e ^(gO, then 工 ^ M. Thus, we have G =< T U {.t} > < M, a contradiction. The theorem is proved. Theorem 1.1.5 (Dedekind Law) Let A,B,C be subgroups of a group G with A C C. Then C 门 AB = 门 B). Proof: Let x e C H AB, then .x = ab e C for some a 6 A, . 6 G B. Since l A C C: b = a~ x E C H B. This implies that .t = ab G A(C 门 B), and hence C D AB C A(C D B). Conversely, let x G A{C 门 B) and write x = ab where ft G A, 6 e C 门 Then it is obvious to see that x G AB. Also, by a e A C C and 6 e C, we have ab G C.