On the Derived Length of Lie Solvable Group Algebras
On the Derived Length of Lie Solvable Group Algebras
On the Derived Length of Lie Solvable Group Algebras
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56 CHAPTER 5<br />
• m ≥ 5<br />
G6 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a−1 〉;<br />
G7 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a−1+2m−3 〉;<br />
G8 = 〈a, b | a2m−2 = 1, b4 = a2m−3 , ab = a−1 〉;<br />
G9 = 〈a, b | a2m−2 = 1, b4 = 1, ba = b−1 〉;<br />
G10 = M2m−1 × C2;<br />
G11 = S2m−1 × C2;<br />
G12 = 〈a, b, c | a 2m−2<br />
G13 = 〈a, b, c | a 2m−2<br />
G14 = 〈a, b, c | a 2m−2<br />
G15 = 〈a, b, c | a 2m−2<br />
G16 = 〈a, b, c | a 2m−2<br />
G17 = 〈a, b, c | a 2m−2<br />
G18 = 〈a, b, c | a 2m−2<br />
= 1, b2 = 1, c2 = 1, ab = ba, ac = a−1 , bc = a2m−3 b〉;<br />
= 1, b2 = 1, c2 = 1, ab = ba, ac = a−1b, bc = cb〉;<br />
= 1, b2 = 1, c2 = a2m−3 , ab = ba, ac = a−1b, bc = cb〉;<br />
= 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = a−1+2m−3 , bc = cb〉;<br />
= 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = a−1+2m−3 , bc = a2m−3 b〉;<br />
= 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = ab, bc = cb〉;<br />
= 1, b2 = 1, c2 = b, ab = a1+2m−3 , ac = a−1b〉; • m ≥ 6<br />
G19 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a1+2m−4 〉;<br />
G20 = 〈a, b | a2m−2 = 1, b4 = 1, ab = a−1+2m−4 〉;<br />
G21 = 〈a, b | a2m−2 = 1, a2m−3 = b4 , a−1ba = b−1 〉;<br />
G22 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = ba, ac = a1+2m−4 b, bc = a2m−3 b〉;<br />
G23 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = ba, ac = a−1+2m−4 b, bc = a2m−3 b〉;<br />
G24 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = a1+2m−3 , ac = a−1+2m−4 b, bc = cb〉;<br />
G25 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = a2m−3 , ab = a1+2m−3 , ac = a−1+2m−4 b, bc =<br />
cb〉;<br />
• G26 = 〈a, b, c | a 8 = 1, b 2 = 1, c 2 = a 4 , a b = a 5 , a c = ab, bc = cb〉,<br />
where for m ≥ 3<br />
and for m ≥ 4<br />
D2m = 〈a, b | a2m−1<br />
Q2m = 〈a, b | a2m−1<br />
= 1, b 2 = 1, a b = a −1 〉;<br />
= 1, b 2 = a 2m−2<br />
, a b = a −1 〉;<br />
S2m = 〈a, b | a2m−1 = 1, b 2 = 1, a b = a −1+2m−2<br />
〉;<br />
M2m = 〈a, b | a2m−1 = 1, b 2 = 1, a b = a 1+2m−2<br />
〉.<br />
The group algebras <strong>of</strong> Gi have been examined by several authors,<br />
for example V. Bódi [9]. Our results enable us to determine <strong>the</strong> derived<br />
length <strong>of</strong> F Gi over a field F <strong>of</strong> characteristic two. The claim is <strong>the</strong><br />
following:<br />
⎧<br />
⎪⎨ 2, if ei<strong>the</strong>r i ∈ {2, 3} and m = 4 or i ∈ {1, 4, 5, 9, 10};<br />
dlL(F Gi) = 4, if i ∈ {15, 16, 18, 20, 24, 25} and m > 5;<br />
⎪⎩<br />
3, o<strong>the</strong>rwise.