On the Derived Length of Lie Solvable Group Algebras

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48 CHAPTER 4
Chapter 5 Group algebras of Lie derived length three The group algebras of Lie derived length two are described in [19] by F. Levin and G. Rosenberger, but which have Lie derived length three are known only for characteristic not less than seven by M. Sahai [24]. The full characterization of these seems to be a difficult problem. A partial solution can be found here for the case when the commutator subgroup of the basic group is cyclic. 5.1 Preliminaries and the description We will use the following Proposition 5.1.1 (A. Shalev [27]). Let G be a group with commutator subgroup of order p n with n > 0 and let char(F ) = p. Then dlL(F G) ≥ ⌈log 2(p + 1)⌉. First of all, in a special case we give an upper bound on the Lie derived length, which will be useful in the sequel. Theorem 5.1.2. Let G be a group and char(F ) = 2. If H is a subgroup of index two of G whose commutator subgroup H ′ is a finite 49
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On the Derived Length of Lie Solvab
Page 5: Acknowledgements I would like to ex
Page 8 and 9: VIII CONTENTS 6 Lie nilpotency indi
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 58 and 59: 50 CHAPTER 5 2-group, then dlL(F G)
Page 60 and 61: 52 CHAPTER 5 Lemma 5.1.4. Let G be
Page 62 and 63: 54 CHAPTER 5 Case 2: G/C = 〈dC〉
Page 64 and 65: 56 CHAPTER 5 • m ≥ 5 G6 = 〈a,
Page 68 and 69: 60 CHAPTER 6 (see Theorem 2.8 of [2
Page 70 and 71: 62 CHAPTER 6 Proposition 6.1.5 (A.A
Page 72 and 73: 64 CHAPTER 6 d(q+1) = 0, then q ≤
Page 74 and 75: 66 CHAPTER 6 Next we consider separ
Page 76 and 77: 68 CHAPTER 6 which is not possible
Page 80 and 81: 72 Basic facts and notations Let G
Page 82 and 83: 74 significant results on this topi
Page 84 and 85: 76 As a consequence, we get a neces
Page 86 and 87: 78 Lie nilpotency indices of group
Page 89 and 90: Összegzés Bevezetés A múlt szá
Page 91 and 92: ÖSSZEGZÉS 83 Alapfogalmak és jel
Page 93 and 94: ÖSSZEGZÉS 85 csoportalgebrák le
Page 95 and 96: ÖSSZEGZÉS 87 A negyedik fejezetbe
Page 97 and 98: ÖSSZEGZÉS 89 Bizonyítottuk még
Page 99 and 100: Bibliography [1] Bagiński, C. A no
Page 101 and 102: BIBLIOGRAPHY 93 [21] Passi, I. B. S
Page 103 and 104: APPENDIX 95 List of papers of the a
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On the Derived Length of Lie Solvable Group Algebras

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