On the Derived Length of Lie Solvable Group Algebras

Recommendations

Info

8 CHAPTER 1 1.2.2 Lie derived lengths of group algebras M. Sahai [24] proved the relation (1.1) I(G ′ ) 2n −1 ⊆ δ (n) (F G) ⊆ I(G ′ ) 2 n−1 for all n > 0, from which it follows that a group algebra F G is strongly Lie solvable if and only if either G is abelian or the ideal I(G ′ ) is nilpotent, that is G ′ is a finite p-group and char(F ) = p. The description of the Lie solvable group algebras is due to I.B.S. Passi, D.S. Passman and S.K. Sehgal [22]: a group algebra F G is Lie solvable if and only if one of the following conditions holds: (i) G is abelian; (ii) G ′ is a finite pgroup and char(F ) = p; (iii) G has a subgroup of index two whose commutator subgroup is a finite 2-group and char(F ) = 2. Clearly, dlL(F G) and dl L (F G) are equal to one if and only if G is an abelian group. The group algebras of Lie derived length (strong Lie derived length) two, that is the so-called Lie metabelian (strongly Lie metabelian) group algebras were described by F. Levin and G. Rosenberger [19]. For odd characteristic the complete list of the strongly Lie solvable group algebras of strong Lie derived length three can be found in M. Sahai’s paper [24]. Moreover, he also showed that the statements δ [3] (F G) = 0 and δ (3) (F G) = 0 are equivalent, provided that char(F ) ≥ 7. All the other cases the question is still open. In general, we have very little information about the Lie derived length of group algebras. The first and, at the same time, the more significant results on this topic can be found in papers [27] and [29] of A. Shalev. Throughout this part by F G we always mean a strongly Lie solvable group algebra. From (1.1) it follows immediately that (1.2) ⌈log 2(tN(G ′ ) + 1)⌉ ≤ dl L (F G) ≤ ⌈log 2(2tN(G ′ ))⌉ and so dlL(F G) ≤ ⌈log 2(2tN(G ′ ))⌉. Since there is no similarly valuable lower bound on dlL(F G), the computation of its value is more difficult than of the value of dl L (F G).
1.2 ASSOCIATED LIE ALGEBRA OF GROUP ALGEBRAS 9 Applying the method used A. Shalev in the proof of Lemma 2.2 in [27] we have that if G is nilpotent of class two, then dlL(F G) ≤ dl L (F G) = ⌈log 2(tN(G ′ ) + 1)⌉. In particular, if G is an abelian-by-cyclic p-group with p > 2 then dlL(F G) = ⌈log 2(tN(G ′ ) + 1)⌉, as it was stated in [29]. The purpose in the second chapter of this thesis will be to extend these results above to a larger class of groups, namely, it is enough to assume that γ3(G) ⊆ (G ′ ) p , instead of the condition G is of class two. The investigation of the next chapter was motivated by the following result of A. Shalev [27]: if G is nilpotent of class two with cyclic commutator subgroup of order p n , then dlL(F G) = ⌈log 2(p n + 1)⌉. We generalize this result and determine both the Lie derived length and the strong Lie derived length of group algebras in the case when the commutator subgroup of the basic group is cyclic of odd order. For characteristic two, when G is a nilpotent group with (not necessary cyclic) commutator subgroup of order 2 n , in the fourth chapter we obtain the description of the group algebras F G which have the highest possible value of dlL(F G), namely, n + 1. By the inclusion δ [n] (F G) ⊆ δ (n) (F G), a strongly Lie solvable group algebra F G is Lie solvable too and dlL(F G) ≤ dl L (F G). It would be also interesting to know when the equality dlL(F G) = dl L (F G) does hold, but this question is still open. As a consequence, we get a necessary and sufficient condition for dlL(F G) to coincide with dl L (F G), provided that G ′ is cyclic. According to (1.2), ⌈log 2(p + 1)⌉ ≤ dl L (F G), where p is the characteristic of F . The characterization of the group algebras of minimal strong Lie derived length is also a consequence of our result. We finish the study of the Lie derived length with the description of the group algebras F G of Lie derived length three, in the case when G ′ is cyclic and a possibility of the application of this result will be shown. This results are contained in Chapter 5.
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 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 66 and 67:
58 CHAPTER 5
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 78 and 79:
70 CHAPTER 6
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
Page 105:
On the Derived Length of Lie Solvab
show all

On the Derived Length of Lie Solvable Group Algebras

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?