On the Derived Length of Lie Solvable Group Algebras

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VIII CONTENTS 6 Lie nilpotency indices of Lie nilpotent group algebras 59 6.1 Preliminary results . . . . . . . . . . . . . . . . . . . . 59 6.2 Group algebras with almost maximal upper Lie nilpotency index . . . . . . . . . . . . . . . . . . . . . . . . 62 Summary 71 Összegzés (Hungarian summary) 81 Bibliography 91 Appendix 95
Chapter 1 Introduction 1.1 Preliminaries At the beginning of the last century an exciting algebraic construction consisting of a group and a field, appeared in the works of G. Frobenius which he used for the observation of the representations of finite groups. This is the construction that was named group algebra in the ’30s by E. Noether. After this, through decades the group algebras were not so much observed for themselves but for the possibilities of their applications in representation theory and algebraic topology, etc. For example, W. Magnus proved his famous theorem on the lower central series of free groups at that time with the aid of group algebras, which theorem was demonstrated using pure group theoretical methods only after a long time. But in the beginning of the ’60s the stress was laid upon the group algebras of infinite groups as a result of I. Kaplansky’s problems in ring theory. The characterization theorems born in this period have led to newer directions in research. Several survey articles were published which brought the formation of this new branch of science. The study of the ring theoretical properties of group algebras is now at the most advanced level, but significant results are known in their unit groups as well. The unit group of group algebras aroused the researchers’ interest for the first time because of their topological applications, and then again, after the description 1
Page 1: On the Derived Length of Lie Solvab
Page 5: Acknowledgements I would like to ex
Page 10 and 11: 2 CHAPTER 1 of simple groups, as sp
Page 12 and 13: 4 CHAPTER 1 is the so-called augmen
Page 14 and 15: 6 CHAPTER 1 and for units a, b that
Page 16 and 17: 8 CHAPTER 1 1.2.2 Lie derived lengt
Page 18 and 19: 10 CHAPTER 1 1.2.3 Lie nilpotency i
Page 20 and 21: 12 CHAPTER 2 (ii) p = 2 and G ′ i
Page 22 and 23: 14 CHAPTER 2 Theorem 2.2.2. Let G b
Page 24 and 25: 16 CHAPTER 2 According to Lemma 2.2
Page 26 and 27: 18 CHAPTER 2
Page 28 and 29: 20 CHAPTER 3 this bound is always a
Page 30 and 31: 22 CHAPTER 3 and if k is even then
Page 32 and 33: 24 CHAPTER 3 since γ3(G) ⊆ (G
Page 34 and 35: 26 CHAPTER 3 Proof. Evidently, x =
Page 36 and 37: 28 CHAPTER 3 and s > 0 the set gω
Page 38 and 39: 30 CHAPTER 3 apply Lemma 3.2.4 to c
Page 40 and 41: 32 CHAPTER 3 hypothesis and (3.11),
Page 42 and 43: 34 CHAPTER 3 and wl+1 ≡ t (l) w t
Page 44 and 45: 36 CHAPTER 3 3.2.3. Finally, let m
Page 46 and 47: 38 CHAPTER 3 and char(F ) = 19. Sin
Page 48 and 49: 40 CHAPTER 4 for suitable odd l. Th
Page 50 and 51: 42 CHAPTER 4 Lemma 4.1.4. Let G be
Page 52 and 53: 44 CHAPTER 4 Similarly, we can obta
Page 54 and 55: 46 CHAPTER 4 The following question
Page 56 and 57: 48 CHAPTER 4
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50 CHAPTER 5 2-group, then dlL(F G)
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52 CHAPTER 5 Lemma 5.1.4. Let G be
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54 CHAPTER 5 Case 2: G/C = 〈dC〉
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56 CHAPTER 5 • m ≥ 5 G6 = 〈a,
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58 CHAPTER 5
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60 CHAPTER 6 (see Theorem 2.8 of [2
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62 CHAPTER 6 Proposition 6.1.5 (A.A
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64 CHAPTER 6 d(q+1) = 0, then q ≤
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66 CHAPTER 6 Next we consider separ
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68 CHAPTER 6 which is not possible
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70 CHAPTER 6
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72 Basic facts and notations Let G
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74 significant results on this topi
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76 As a consequence, we get a neces
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78 Lie nilpotency indices of group
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Összegzés Bevezetés A múlt szá
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ÖSSZEGZÉS 83 Alapfogalmak és jel
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ÖSSZEGZÉS 85 csoportalgebrák le
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ÖSSZEGZÉS 87 A negyedik fejezetbe
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ÖSSZEGZÉS 89 Bizonyítottuk még
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Bibliography [1] Bagiński, C. A no
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BIBLIOGRAPHY 93 [21] Passi, I. B. S
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APPENDIX 95 List of papers of the a
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On the Derived Length of Lie Solvab
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On the Derived Length of Lie Solvable Group Algebras

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