On the Derived Length of Lie Solvable Group Algebras

Chapter 6
Lie nilpotency indices of Lie
nilpotent group algebras
According to [32], if F G is Lie nilpotent and G ′ has order p n , then
tL(F G) ≤ t L (F G) ≤ p n + 1.
A. Shalev in [25] began to study the question when a Lie nilpotent
group algebra has the maximal upper Lie nilpotency index. The complete
description of such group algebras was given by V. Bódi and E.
Spinelli in [13]. In this chapter we determine the group algebras whose
upper Lie nilpotency index is ‘almost maximal’, that is, it takes the
next highest possible value, namely p n − p + 2, where p n is the order
of the commutator subgroup of the basic group.
6.1 Preliminary results
Let F G be a Lie nilpotent group algebra. We consider a sequence
of subgroups of G, setting
D(m)(G) = G ∩ (1 + F G (m) ), (m ≥ 1).
The subgroup D(m)(G) is called the m-th Lie dimension subgroup of
F G. It is possible to describe the D(m)(G)’s in the following manner
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