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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26 CHAPTER 3<br />
Pro<strong>of</strong>. Evidently, x = (a, b) is <strong>of</strong> order p n , x b = x and x a = x −1 .<br />
Define <strong>the</strong> following three series <strong>of</strong> elements <strong>of</strong> F G inductively by<br />
and for l ≥ 0, by<br />
u0 = a, v0 = b, w0 = b −1 a,<br />
ul+1 = [ul, vl], vl+1 = [ul, wl], wl+1 = [wl, vl].<br />
By induction on l we show for odd l that<br />
(3.9)<br />
ul ≡ t (l)<br />
u ba(x−1 − 1) sl−2 (x − 1) sl−2+1<br />
vl ≡ t (l)<br />
v b−1 (x −1 − 1) sl−2 (x − 1) sl−2+1<br />
wl ≡ t (l)<br />
w a(x −1 − 1) sl−2 (x − 1) sl−2+1<br />
and if l is even <strong>the</strong>n<br />
(3.10)<br />
(mod I(G ′ ) sl−1+2 );<br />
(mod I(G ′ ) sl−1+2 );<br />
(mod I(G ′ ) sl−1+2 ),<br />
ul ≡ t (l)<br />
u a(x−1 − 1) sl−2 (x − 1) sl−2 (mod I(G ′ ) sl−1+2 );<br />
vl ≡ t (l)<br />
v b(x−1 − 1) sl−2+1 (x − 1) sl−2 (mod I(G ′ ) sl−1+2 );<br />
wl ≡ t (l)<br />
w b −1 a(x −1 − 1) sl−2 (x − 1) sl−2 (mod I(G ′ ) sl−1+2 ),<br />
where t (l)<br />
u , t (l)<br />
v , t (l)<br />
w are nonzero elements in <strong>the</strong> field F , and for convenience<br />
let us set sl = s (1)<br />
l with s−1 = 0. Obviously, u1 = [a, b] =<br />
ba(x − 1),<br />
v1 = [a, b −1 a] = b −1 (x −1 ) a − 1 = b −1 (x − 1),<br />
and similarly, w1 = [b −1 a, b] = a(x − 1). Therefore (3.9) holds for<br />
l = 1. Now, assume that (3.9) is true for some odd l. Then by (3.8),<br />
ul+1<br />
≡ t (l)<br />
u t (l)<br />
v [ba(x −1 − 1) sl−2 sl−2+1 −1 −1 sl−2 sl−2+1<br />
(x − 1) , b (x − 1) (x − 1) ]<br />
≡ t (l)<br />
u t (l)<br />
<br />
−1 −1 2sl−2 2sl−2+2<br />
v bab (x − 1) (x − 1)<br />
− a(x −1 − 1) 2sl−2+1 2sl−2+1<br />
(x − 1) <br />
≡ (−2)t (l)<br />
u t(l)<br />
v a(x−1 − 1) sl−1 (x − 1) sl−1 (mod I(G ′ ) sl+2 );