On the Derived Length of Lie Solvable Group Algebras

4.2 THE DESCRIPTION AND SOME CONSEQUENCES 45
Lemma 4.1.5. Let G be a nilpotent group with commutator subgroup
G ′ = 〈x | x2n = 1〉, char(F ) = p and assume that n ≥ 3 and G has
nilpotency class at most n. Then
dlL(F G) = ⌈log 2(p n + 1)⌉.
Proof. Let us repeat the proof of Theorem 3.1.2 with the following
modification: to obtain the inclusion
b l a m , (x − 1) 2 k −1 , (x − 1) 2k −1 , b s a t ∈ I(G ′ ) 2k +1
let us apply Lemma 4.1.1(ii), in view of the fact that now b l a m , b s a t ∈
Gβ by Lemma 4.1.2.
4.2 The description and some consequences
The main theorem of this chapter can be stated as follows:
Theorem 4.2.1. Let G be a nilpotent group with commutator subgroup
of order 2 n and let F be a field of characteristic two. Then dlL(F G) =
n + 1 if and only if one of the following conditions holds:
(i) G ′ is the noncyclic group of order 4 and γ3(G) = 1;
(ii) G ′ is cyclic of order less than 8;
(iii) G ′ is cyclic, n ≥ 3 and G has nilpotency class at most n.
Proof. Assume first that G ′ is cyclic. For p = 2 and n < 3 the statement
is a consequence of Theorem 2.2.2 and Proposition 2.1.2. In the
other cases Lemma 4.1.4 and Lemma 4.1.5 state the required result.
Now, assume that G ′ is noncyclic and δ [n] (F G) = 0. We know that F G
is Lie nilpotent, and as we have already seen, δ [n] (F G) ⊆ (F G) (2n ) .
Thus (F G) (2n ) = 0 and Proposition 6.1.1 (see in Chapter 6) states
that G ′ = C2 × C2 and γ3(G) = 1. Conversely, if G ′ = C2 × C2
then tN(G ′ ) = 3 and dlL(F G) ≤ ⌈log 2 (2 · 3)⌉ = 3. Furthermore,
when γ3(G) = 1, Proposition 2.1.2 says that dlL(F G) = 2. Therefore
dlL(F G) = 3 and the theorem is proved.