On the Derived Length of Lie Solvable Group Algebras

Chapter 2
An extension of a result of A.
Shalev
As it was proved in [27, 29] by A. Shalev, if G is a nilpotent group
of class two and char(F ) = p, then dlL(F G) ≤ ⌈log 2(tN(G ′ ) + 1)⌉ and,
in particular, if G is an abelian-by-cyclic p-group with p > 2 then
the equality holds. Our goal is to extend this result to groups G for
which γ3(G) ⊆ (G ′ ) p holds. We note that these groups are nilpotent,
according to a result of A.G.R. Stewart [33].
2.1 Preliminary results
We will refer to the following statements in the proofs and the
examples.
Proposition 2.1.1 (M. Sahai [24]). For all n ≥ 1
I(G ′ ) 2n −1 ⊆ δ (n) (F G) ⊆ I(G ′ ) 2 n−1
.
Proposition 2.1.2 (F. Levin and G. Rosenberger [19]). The group
algebra F G is Lie metabelian if and only if one of the following statements
is satisfied:
(i) p = 3 and G ′ is central of order 3;
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