56 CHAPTER 5 • m ≥ 5 G6 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a−1 〉; G7 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a−1+2m−3 〉; G8 = 〈a, b | a2m−2 = 1, b4 = a2m−3 , ab = a−1 〉; G9 = 〈a, b | a2m−2 = 1, b4 = 1, ba = b−1 〉; G10 = M2m−1 × C2; G11 = S2m−1 × C2; G12 = 〈a, b, c | a 2m−2 G13 = 〈a, b, c | a 2m−2 G14 = 〈a, b, c | a 2m−2 G15 = 〈a, b, c | a 2m−2 G16 = 〈a, b, c | a 2m−2 G17 = 〈a, b, c | a 2m−2 G18 = 〈a, b, c | a 2m−2 = 1, b2 = 1, c2 = 1, ab = ba, ac = a−1 , bc = a2m−3 b〉; = 1, b2 = 1, c2 = 1, ab = ba, ac = a−1b, bc = cb〉; = 1, b2 = 1, c2 = a2m−3 , ab = ba, ac = a−1b, bc = cb〉; = 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = a−1+2m−3 , bc = cb〉; = 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = a−1+2m−3 , bc = a2m−3 b〉; = 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = ab, bc = cb〉; = 1, b2 = 1, c2 = b, ab = a1+2m−3 , ac = a−1b〉; • m ≥ 6 G19 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a1+2m−4 〉; G20 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a−1+2m−4 〉; G21 = 〈a, b | a2m−2 = 1, a2m−3 = b4 , a−1ba = b−1 〉; G22 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = ba, ac = a1+2m−4 b, bc = a2m−3 b〉; G23 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = ba, ac = a−1+2m−4 b, bc = a2m−3 b〉; G24 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = a−1+2m−4 b, bc = cb〉; G25 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = a2m−3 , ab = a1+2m−3 , ac = a−1+2m−4 b, bc = cb〉; • G26 = 〈a, b, c | a 8 = 1, b 2 = 1, c 2 = a 4 , a b = a 5 , a c = ab, bc = cb〉, where for m ≥ 3 and for m ≥ 4 D2m = 〈a, b | a2m−1 Q2m = 〈a, b | a2m−1 = 1, b 2 = 1, a b = a −1 〉; = 1, b 2 = a 2m−2 , a b = a −1 〉; S2m = 〈a, b | a2m−1 = 1, b 2 = 1, a b = a −1+2m−2 〉; M2m = 〈a, b | a2m−1 = 1, b 2 = 1, a b = a 1+2m−2 〉. The group algebras <strong>of</strong> Gi have been examined by several authors, for example V. Bódi [9]. Our results enable us to determine <strong>the</strong> derived length <strong>of</strong> F Gi over a field F <strong>of</strong> characteristic two. The claim is <strong>the</strong> following: ⎧ ⎪⎨ 2, if ei<strong>the</strong>r i ∈ {2, 3} and m = 4 or i ∈ {1, 4, 5, 9, 10}; dlL(F Gi) = 4, if i ∈ {15, 16, 18, 20, 24, 25} and m > 5; ⎪⎩ 3, o<strong>the</strong>rwise.
A POSSIBILITY OF APPLICATION 57 Indeed, apart from <strong>the</strong> cases i ∈ {17, 26}, G ′ i is cyclic. If ei<strong>the</strong>r i ∈ {2, 3} and m = 4 or i ∈ {1, 4, 5, 9, 10} <strong>the</strong>n G ′ i = C2 and <strong>the</strong> statement follows from e. g. Theorem 3.3.1. Fur<strong>the</strong>rmore, if i ∈ {15, 16, 18, 20, 24, 25} and m > 5 <strong>the</strong>n Gi has a normal subgroup with commutator subgroup <strong>of</strong> order two. Using Theorem 5.1.2 it follows that dlL(F Gi) ≤ 4. To prove <strong>the</strong> converse inequality we can use Theorem 5.1.6. In all <strong>the</strong> o<strong>the</strong>r cases ei<strong>the</strong>r G ′ i is <strong>of</strong> order 4 or Gi has an abelian subgroup <strong>of</strong> index two. Then Theorem 5.1.6 guaranteers <strong>the</strong> statement. Finally, G ′ 17 ∼ = G ′ 26 ∼ = C2 × C2 and we can apply Proposition 2.1.2 to compute <strong>the</strong> derived length.
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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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