On the Derived Length of Lie Solvable Group Algebras

10 CHAPTER 1
1.2.3 Lie nilpotency indices of group algebras
The object of the sixth chapter is the investigation of the Lie nilpotency
indices of Lie nilpotent group algebras. For the noncommutative
modular group algebra F G the next theorem from A.A. Bódi and I.I.
Khripta [7] is well-known: The following statements are equivalent:
(i) F G is Lie nilpotent; (ii) F G is upper Lie nilpotent; (iii) G is a
nilpotent group whose commutator subgroup is a finite p-group and
char(F ) = p.
Since (F G) [n] ⊆ (F G) (n) for all n, tL(F G) ≤ t L (F G). Moreover,
for characteristic p > 3 the equality is also guaranteed by a result
of A.K. Bhandari and I.B.S. Passi [2], but the problem is still open
otherwise.
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, but the
complete description of such group algebras was given by V. Bódi
and E. Spinelli in [13]. Joining this research 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.