On the Derived Length of Lie Solvable Group Algebras

Chapter 5
Group algebras of Lie derived
length three
The group algebras of Lie derived length two are described in [19]
by F. Levin and G. Rosenberger, but which have Lie derived length
three are known only for characteristic not less than seven by M. Sahai
[24]. The full characterization of these seems to be a difficult problem.
A partial solution can be found here for the case when the commutator
subgroup of the basic group is cyclic.
5.1 Preliminaries and the description
We will use the following
Proposition 5.1.1 (A. Shalev [27]). Let G be a group with commutator
subgroup of order p n with n > 0 and let char(F ) = p. Then
dlL(F G) ≥ ⌈log 2(p + 1)⌉.
First of all, in a special case we give an upper bound on the Lie
derived length, which will be useful in the sequel.
Theorem 5.1.2. Let G be a group and char(F ) = 2. If H is a subgroup
of index two of G whose commutator subgroup H ′ is a finite
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