On the Derived Length of Lie Solvable Group Algebras

78
Lie nilpotency indices of group algebras
In the sixth chapter we study the Lie nilpotency indices of Lie
nilpotent group algebras. Let (F G) [1] = F G and for n > 1 let (F G) [n]
be the ideal of F G generated by all the Lie commutators [x1, . . . , xn]
with x1, . . . , xn ∈ F G. Then the ideal (F G) [n] is the n-th lower Lie
power and the series
F G = (F G) [1] =⊇ (F G) [2] ⊇ · · · ⊇ (F G) [n] ⊇ · · ·
is called the lower Lie power series of the group algebra F G.
By induction, we define the n-th upper Lie power (F G) (n) of F G
as the ideal generated by all the Lie commutators [x, y], where x ∈
(F G) (n−1) , y ∈ F G and (F G) (1) = F G. The series
F G = (F G) (1) ⊇ (F G) (2) ⊇ · · · ⊇ (F G) (n) ⊇ · · ·
is the upper Lie power series of F G.
The group algebra F G is called Lie nilpotent if there exists n such
that (F G) [n] = 0 and the least integer of this kind is called the Lie
nilpotency index of F G and it is denoted by tL(F G). Similarly, F G
is said to be upper Lie nilpotent and its upper Lie nilpotency index is
t L (F G) = m if (F G) (m) = 0 but (F G) (m−1) = 0. 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.
According to [32], if F G is Lie nilpotent and G ′ has order p n , then
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]. 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.