On the Derived Length of Lie Solvable Group Algebras

22 CHAPTER 3
and if k is even then
(3.5)
uk ≡ ±a(x − 1) 2k −1
vk ≡ ±b(x − 1) 2k −1
wk ≡ ±b −1 a −1 (x − 1) 2k −1
(mod I(G ′ ) 2k
);
(mod I(G ′ ) 2k
);
(mod I(G ′ ) 2k
).
Evidently, u1 = [a, b] = ba(x − 1), and from (3.1) it follows that
v1 = [a, b −1 a −1 ] ≡ −b −1 (x − 1) (mod I(G ′ ) 2 ),
and w1 = [b −1 a −1 , b] ≡ −a −1 (x − 1) (mod I(G ′ ) 2 ). Therefore (3.4)
holds for k = 1.
Now, assume that (3.4) is true for some odd k. According to (3.3) the
congruences
uk+1 = ±fk(1, 1, −1, 0, uk ′ , vk ′ )
≡ ±(−1)a(x − 1) 2k+1 −1
vk+1 = ±fk(1, 1, 0, −1, uk ′ , vk ′ )
≡ ±b(x − 1) 2k+1 −1
wk+1 = ±fk(0, −1, −1, 0, uk ′ , vk ′ )
≡ ±b −1 a −1 (x − 1) 2k+1 −1
(mod I(G ′ ) 2k+1
);
(mod I(G ′ ) 2k+1
);
(mod I(G ′ ) 2k+1
)
hold, where uk ′ , vk ′ , wk ′ are suitable elements from I(G ′ ) 2k.
Similarly,
supposing the truth of (3.5) for some even k we see
uk+1 = ±fk(0, 1, 1, 0, uk ′ , vk ′ )
≡ ±ba(x − 1) 2k+1 −1
vk+1 = ±fk(0, 1, −1, −1, uk ′ , vk ′ )
≡ ±(−1)b −1 (x − 1) 2k+1 −1
wk+1 = ±fk(−1, −1, 1, 0, uk ′ , vk ′ )
≡ ±(−1)a −1 (x − 1) 2k+1 −1
So, (3.4) and (3.5) are valid for any k > 0.
(mod I(G ′ ) 2k+1
);
(mod I(G ′ ) 2k+1
);
(mod I(G ′ ) 2k+1
).