On the Derived Length of Lie Solvable Group Algebras

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92 BIBLIOGRAPHY [10] Bovdi, V. Modular group algebras with almost maximal Lie nilpotency indices. II. to appear in Scientiae Mathematicae Japonicae (SCMJ) [11] Bovdi, V.; Dokuchaev, M. Group algebras whose involutory units commute. Algebra Colloq. 9 (2002), no. 1, 49–64. [12] Bovdi, V.; Konovalov, A. B.; Rossmanith, A. R.; Schneider, Cs. Laguna — Lie AlGebras and UNits of group Algebras. (http://ukrgap.exponenta.ru/ laguna.htm), Version 3.0, 2003. [13] Bovdi, V.; Spinelli, E. Modular group algebras with maximal Lie nilpotency indices. Publ. Math. Debrecen 65 (2004), no. 1-2, 243– 252. [14] Brauer, R. Zur Darstellungstheorie der Gruppen endlicher Ordnung. (German) Math. Z. 63 (1956), 406–444. [15] Huppert, B. Endliche Gruppen. I. (German) Die Grundlehren der Mathematischen Wissenschaften, Band 134 Springer-Verlag, Berlin-New York 1967 xii+793 pp. [16] Jennings, S. A. The structure of the group ring of a p-group over a modular field. Trans. Amer. Math. Soc. 50, (1941). 175–185. [17] Kurdics, J. Engel properties of group algebras. I. Publ. Math. Debrecen 49 (1996), no. 1-2, 183–192. [18] Kurdics, J. Engel properties of group algebras. II. Ring theory (Miskolc, 1996). J. Pure Appl. Algebra 133 (1998), no. 1-2, 179– 196. [19] Levin, F.; Rosenberger, G. Lie metabelian group rings. Group and semigroup rings (Johannesburg, 1985), 153–161, North-Holland Math. Stud., 126, North-Holland, Amsterdam, 1986. [20] Ninomiya, Y. Finite p-groups with cyclic subgroups of index p 2 . Math. J. Okayama Univ. 36 (1994), 1–21 (1995).
BIBLIOGRAPHY 93 [21] Passi, I. B. S. Group rings and their augmentation ideals. Lecture Notes in Mathematics, 715. Springer, Berlin, 1979. vi+137 pp. [22] Passi, I. B. S.; Passman, D. S.; Sehgal, S. K. Lie solvable group rings. Canad. J. Math. 25 (1973), 748–757. [23] Rossmanith, R. Lie centre-by-metabelian group algebras in even characteristic. I, II. Israel J. Math. 115 (2000), 51–75, 77–99. [24] Sahai, M. Lie solvable group algebras of derived length three. Publ. Mat. 39 (1995), no. 2, 233–240. [25] Shalev, A. Applications of dimension and Lie dimension subgroups to modular group algebras. Ring theory 1989 (Ramat Gan and Jerusalem, 1988/1989), 84–95, Israel Math. Conf. Proc., 1, Weizmann, Jerusalem, 1989. [26] Shalev, A. Lie dimension subgroups, Lie nilpotency indices, and the exponent of the group of normalized units. J. London Math. Soc. (2) 43 (1991), no. 1, 23–36. [27] Shalev, A. The derived length of Lie soluble group rings. I. J. Pure Appl. Algebra 78 (1992), no. 3, 291–300. [28] Shalev, A. The nilpotency class of the unit group of a modular group algebra. III. Arch. Math. (Basel) 60 (1993), no. 2, 136–145. [29] Shalev, A. The derived length of Lie soluble group rings. II. J. London Math. Soc. (2) 49 (1994), no. 1, 93–99. [30] Sharma, R. K.; Srivastava, J. B. Lie ideals in group rings. J. Pure Appl. Algebra 63 (1990), no. 1, 67–80. [31] Sharma, R. K.; Srivastava, J. B. Lie centrally metabelian group rings. J. Algebra 151 (1992), no. 2, 476–486. [32] Sharma, R. K.; Bist, V. A note on Lie nilpotent group rings. Bull. Austral. Math. Soc. 45 (1992), no. 3, 503–506.
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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
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18 CHAPTER 2
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20 CHAPTER 3 this bound is always a
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22 CHAPTER 3 and if k is even then
Page 32 and 33:
24 CHAPTER 3 since γ3(G) ⊆ (G
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26 CHAPTER 3 Proof. Evidently, x =
Page 36 and 37:
28 CHAPTER 3 and s > 0 the set gω
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30 CHAPTER 3 apply Lemma 3.2.4 to c
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32 CHAPTER 3 hypothesis and (3.11),
Page 42 and 43:
34 CHAPTER 3 and wl+1 ≡ t (l) w t
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36 CHAPTER 3 3.2.3. Finally, let m
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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 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: Bibliography [1] Bagiński, C. A no
Page 103 and 104: APPENDIX 95 List of papers of the a
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