On the Derived Length of Lie Solvable Group Algebras

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68 CHAPTER 6 which is not possible either. Case 2.c: Let γ3(G) = 〈a2j〉 × 〈b〉 with 1 ≤ j ≤ n − 2. It is easy to check that this case is similar to the last subcase of the case 2.b. So the proof is complete. Lemma 6.2.3. Let G be a nilpotent group with commutator subgroup of order 3 n and let char(F ) = 3. If t L (F G) = 3 n − 1 then G is of class 3, G ′ ∼ = C3 × C3 and γ3(G) ∼ = C3. Proof. Let t L (F G) = 3 n − 1. Then either (iii) or (iv) of Lemma 6.2.1 holds. By (iv) of Lemma 6.2.1 it yields d (3 i +1) = 1, d(3 n−1 ) = 1, d(j) = 0, where 0 ≤ i ≤ n − 2, j = 3 i + 1, j = 3 n−1 and j > 1. The subgroup H = D (3 n−1 )(G) is central of order 3 and from (6.2), as we already proved at the beginning of the proof of Lemma 6.2.2, we have D(m+1)(G)/H = D(m+1)(G/H). It follows that d (3i +1) = 1 for 0 ≤ i ≤ n − 2, d(j) = 0 for j = 3i + 1 and j > 1, where 3d (k) = [D(k)(G/H) : D(k+1)(G/H)]. Clearly, γ2(G/H) has order = 3n−1 and tL (F [G/H]) = 3n−1 + 1. By Propositions 6.1.1 and 6.1.4 we have that γ2(G/H) is a cyclic 3group. Then γ2(G) is abelian, so it is isomorphic to either C3n−1 × C3 or C3n. By Proposition 6.1.1, in the last case F G has upper and lower maximal Lie nilpotency index. Therefore we can assume that γ2(G) ∼ = C3n−1 ×C3 and n ≥ 3. Since exp(γ2(G)) = 3n−1 and 3n−2 divides 2 · 3n−2 , by (iii) of Lemma 6.2.1 and by (ii) of Proposition 6.1.3 we obtain that D (3n−2 +1)(G) = 〈1〉 and so D (3n−1 )(G) = 〈1〉 in contradiction with (iii) of Lemma 6.2.1. It follows that γ2(G) ∼ = C3 × C3. Suppose that γ2(G) ⊆ ζ(G). By (6.1) we get D(3)(G) = 〈1〉 and so d(2) = 2, which is in contradiction with (iii) of Lemma 6.2.1. Finally, the case (iii) of Lemma 6.2.1 follows from the previous case. The proof is done.
GROUP ALGEBRAS WITH ALMOST MAXIMAL... 69 Finally we can state the main theorem of this chapter. Theorem 6.2.4. Let F G be a Lie nilpotent group algebra over a field F of positive characteristic p. Then F G has upper almost maximal Lie nilpotency index if and only if one of the following conditions holds: (i) p = 2, G is of class 2 and G ′ is noncyclic of order 4; (ii) p = 2, G is of class 4, G ′ = C4 × C2 and γ3(G) = C2 × C2; (iii) p = 2, G is of class 4 and G ′ is elementary abelian of order 8; (iv) p = 3, G is of class 3 and G ′ is elementary abelian of order 9. Proof. Assuming that F G has upper almost maximal Lie nilpotency index, the statement is a consequence of the previous lemmas. The converse is obvious by (6.3). We mention that during the preparation of this thesis V. Bódi [10] proved the following statement: F G has upper almost maximal Lie nilpotency index if and only if F G has lower almost maximal Lie nilpotency index.
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On the Derived Length of Lie Solvab
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Acknowledgements I would like to ex
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VIII CONTENTS 6 Lie nilpotency indi
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2 CHAPTER 1 of simple groups, as sp
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4 CHAPTER 1 is the so-called augmen
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6 CHAPTER 1 and for units a, b that
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8 CHAPTER 1 1.2.2 Lie derived lengt
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10 CHAPTER 1 1.2.3 Lie nilpotency i
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12 CHAPTER 2 (ii) p = 2 and G ′ i
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14 CHAPTER 2 Theorem 2.2.2. Let G b
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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 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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