On the Derived Length of Lie Solvable Group Algebras

Chapter 3
Lie derived lengths of group
algebras of groups with cyclic
commutator subgroup
In this chapter we determine the Lie derived length and the strong
Lie derived length of group algebras in the case when the commutator
subgroup of the basic group is cyclic of odd order.
We distinguish two cases according as the basic group is nilpotent
or not.
3.1 The basic group is nilpotent
A. Shalev proved the following
Proposition 3.1.1 (A. Shalev [27]). Let G be a nilpotent group of
class two with cyclic commutator subgroup of order p n and let
char(F ) = p. Then
dlL(F G) = ⌈log 2(p n + 1)⌉.
Evidently, when G ′ is cyclic and G is nilpotent, the condition
γ3(G) ⊆ (G ′ ) p holds, therefore Theorem 2.2.2 ensures that the integer
⌈log 2(p n + 1)⌉ is an upper bound on dlL(F G). It is shown here that
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