On the Derived Length of Lie Solvable Group Algebras

SUMMARY 73
Lie commutator by induction as
[x1, x2, . . . , xn] =
[x1, x2, . . . , xn−1], xn .
Lie derived lengths of group algebras
Define the Lie derived series and the strong Lie derived series
of the group algebra F G respectively, as follows: let δ [0] (F G) =
δ (0) (F G) = F G and
δ [n+1] (F G) = δ [n] (F G), δ [n] (F G) ,
δ (n+1) (F G) = δ (n) (F G), δ (n) (F G) F G.
We say that F G is Lie solvable if there exists m ∈ N such that
δ [m] (F G) = 0 and the number dlL(F G) = min{m ∈ N | δ [m] (F G) = 0}
is called the Lie derived length of F G. Similarly, the group algebra
F G is said to be strongly Lie solvable of derived length dl L (F G) = m
if δ (m) (F G) = 0 and δ (m−1) (F G) = 0.
According to the inclusion δ [n] (F G) ⊆ δ (n) (F G), a strongly Lie
solvable group algebra F G is Lie solvable too and dlL(F G) ≤ dl L (F G).
It would be also interesting to know when the equality dlL(F G) =
dl L (F G) does hold, but this question is still open.
M. Sahai [24] proved the relation
(∗) I(G ′ ) 2n −1 ⊆ δ (n) (F G) ⊆ I(G ′ ) 2 n−1
for all n > 0,
from which it follows that a group algebra F G is strongly Lie solvable
if and only if either G is abelian or the ideal I(G ′ ) is nilpotent, that
is G ′ is a finite p-group and char(F ) = p. The description of the Lie
solvable group algebras is due to I.B.S. Passi, D.S. Passman and S.K.
Sehgal [22]: a group algebra F G is Lie solvable if and only if one of
the following conditions holds: (i) G is abelian; (ii) G ′ is a finite pgroup
and char(F ) = p; (iii) G has a subgroup of index two whose
commutator subgroup is a finite 2-group and char(F ) = 2.
In general, we have very little information about the Lie derived
length of group algebras. The first and, at the same time, the more