On the Derived Length of Lie Solvable Group Algebras

12 CHAPTER 2
(ii) p = 2 and G ′ is central elementary abelian subgroup of order
dividing 4.
Moreover, dlL(F G) = 2 if and only if dl L (F G) = 2.
Proposition 2.1.3 (M. Sahai [24]). Let G be a group and let F be a
field of characteristic p > 2. Then δ (3) (F G) = 0 if and only if one of
the following conditions holds:
(i) G is abelian;
(ii) p = 7, G ′ = C7 and γ3(G) = 1;
(iii) p = 5, G ′ = C5 and either γ3(G) = 1 or γn(G) = G ′ for all
n ≥ 3 with x g = x −1 for all x ∈ G ′ and g ∈ CG(G ′ );
(iv) p = 3 and G ′ is a group of one of the following types:
a) G ′ = C3;
b) G ′ = C3 ×C3 and either γ3(G) = 1 or γ3(G) = C3, γ4(G) =
1 or γn(G) = G ′ for all n ≥ 3 with x g = x −1 for all x ∈ G ′
and g ∈ CG(G ′ );
c) G ′ = C3 × C3 × C3 and γ3(G) = 1.
Proposition 2.1.4 (A.A. Bódi and J. Kurdics [8]). Let G be a nilpotent
group whose commutator subgroup is a finite p-group and let
char(F ) = p. If γ3(G) ⊆ (G ′ ) p then (F G) (n) = I(G ′ ) n−1 for all n ≥ 2.
2.2 The generalized result
We need the following lemma.
Lemma 2.2.1. Let G be a group with commutator subgroup of order
p n , char(F ) = p and assume that γ3(G) ⊆ (G ′ ) p . Then for all m ≥ 1
ω m (F G ′ ), ω(F G) ⊆ I(G ′ ) m+p−1 .
Moreover, if G ′ is abelian, then for all m, k ≥ 1
I(G ′ ) m , I(G ′ ) k ⊆ I(G ′ ) m+k+1 .