On the Derived Length of Lie Solvable Group Algebras

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 of Gi have been examined by several authors,
for example V. Bódi [9]. Our results enable us to determine the derived
length of F Gi over a field F of characteristic two. The claim is the
following:
⎧
⎪⎨ 2, if either 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, otherwise.