On the Derived Length of Lie Solvable Group Algebras

Recommendations

Info

78 Lie nilpotency indices of group algebras In the sixth chapter we study the Lie nilpotency indices of Lie nilpotent group algebras. Let (F G) [1] = F G and for n > 1 let (F G) [n] be the ideal of F G generated by all the Lie commutators [x1, . . . , xn] with x1, . . . , xn ∈ F G. Then the ideal (F G) [n] is the n-th lower Lie power and the series F G = (F G) [1] =⊇ (F G) [2] ⊇ · · · ⊇ (F G) [n] ⊇ · · · is called the lower Lie power series of the group algebra F G. By induction, we define the n-th upper Lie power (F G) (n) of F G as the ideal generated by all the Lie commutators [x, y], where x ∈ (F G) (n−1) , y ∈ F G and (F G) (1) = F G. The series F G = (F G) (1) ⊇ (F G) (2) ⊇ · · · ⊇ (F G) (n) ⊇ · · · is the upper Lie power series of F G. The group algebra F G is called Lie nilpotent if there exists n such that (F G) [n] = 0 and the least integer of this kind is called the Lie nilpotency index of F G and it is denoted by tL(F G). Similarly, F G is said to be upper Lie nilpotent and its upper Lie nilpotency index is t L (F G) = m if (F G) (m) = 0 but (F G) (m−1) = 0. For the noncommutative modular group algebra F G the next theorem from A.A. Bódi and I.I. Khripta [7] is well-known: The following statements are equivalent: (i) F G is Lie nilpotent; (ii) F G is upper Lie nilpotent; (iii) G is a nilpotent group whose commutator subgroup is a finite p-group and char(F ) = p. According to [32], if F G is Lie nilpotent and G ′ has order p n , then t L (F G) ≤ p n + 1. A. Shalev in [25] began to study the question when a Lie nilpotent group algebra has the maximal upper Lie nilpotency index. The complete description of such group algebras was given by V. Bódi and E. Spinelli in [13]. Joining this research we determine the group algebras whose upper Lie nilpotency index is ‘almost maximal’, that is, it takes the next highest possible value, namely p n −p+2, where p n is the order of the commutator subgroup of the basic group.
SUMMARY 79 Theorem. 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 ′ = C2 × C2; (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 ′ = C2 × C2 × C2; (iv) p = 3, G is of class 3 and G ′ = C3 × C3.
Page 1:
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 56 and 57: 48 CHAPTER 4
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 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
Page 105: On the Derived Length of Lie Solvab
show all

On the Derived Length of Lie Solvable Group Algebras

Create successful ePaper yourself

Delete template?

Save as template?