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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14 CHAPTER 2<br />
Theorem 2.2.2. Let G be a nilpotent group whose commutator subgroup<br />
is a finite p-group, char(F ) = p and assume that γ3(G) ⊆ (G ′ ) p .<br />
Then<br />
dlL(F G) ≤ dl L (F G) = ⌈log 2(tN(G ′ ) + 1)⌉,<br />
and if G is an abelian-by-cyclic p-group with p > 2 <strong>the</strong>n<br />
dlL(F G) = dl L (F G) = ⌈log 2 tN(G ′ ) + 1⌉.<br />
Pro<strong>of</strong>. Recall that δ (n) (F G) ⊆ (F G) (2n ) for all n ≥ 0. For n ≥ 1<br />
Proposition 2.1.1 yields that<br />
I(G ′ ) 2n −1 ⊆ δ (n) (F G)<br />
and since γ3(G) ⊆ (G ′ ) p , by Proposition 2.1.4, we have<br />
Combining this facts we get<br />
(F G) (2n ) = I(G ′ ) 2 n −1 .<br />
I(G ′ ) 2n −1 ⊆ δ (n) (F G) ⊆ (F G) (2 n ) = I(G ′ ) 2 n −1 .<br />
Now it is easy to see that δ (n) (F G) = 0 if and only if 2n − 1 ≥ tN(G ′ ),<br />
<strong>the</strong>refore n ≥ log2(tN(G ′ ) + 1), which implies <strong>the</strong> statement.<br />
Let now G be an abelian-by-cyclic p-group with p > 2. Then<br />
G = 〈A, x〉 for some abelian normal subgroup A <strong>of</strong> G and x ∈ G.<br />
Clearly, G ′ = (A, x) is abelian and all z ∈ G ′ can be written in <strong>the</strong><br />
form z = (a1, x) · · · (as, x) for some a1, . . . , as ∈ A.<br />
We are going to show that if c ∈ A, z1, z2, . . . , z2n−1 ∈ G ′ and j is<br />
not divisible by p, <strong>the</strong>n <strong>the</strong>re exists ϱ ∈ I(G ′ ) 2n such that<br />
x j c(1 − z1)(1 − z2) · · · (1 − z2 n −1) + ϱ ∈ δ [n] (F G).<br />
We use induction on n. Let first n = 1 and let us choose an integer<br />
k such that 2k ≡ j modulo <strong>the</strong> order <strong>of</strong> x. Then G ′ = (A, x k ) and<br />
z1 = (a1, x k ) · · · (as, x k ) for some a1, . . . , as ∈ A. Applying <strong>the</strong> identity<br />
1 − αβ = (1 − α) + (1 − β) − (1 − α)(1 − β)