On the Derived Length of Lie Solvable Group Algebras

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