On the Derived Length of Lie Solvable Group Algebras

8 CHAPTER 1
1.2.2 Lie derived lengths of group algebras
M. Sahai [24] proved the relation
(1.1) 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.
Clearly, dlL(F G) and dl L (F G) are equal to one if and only if G is
an abelian group. The group algebras of Lie derived length (strong Lie
derived length) two, that is the so-called Lie metabelian (strongly Lie
metabelian) group algebras were described by F. Levin and G. Rosenberger
[19]. For odd characteristic the complete list of the strongly
Lie solvable group algebras of strong Lie derived length three can be
found in M. Sahai’s paper [24]. Moreover, he also showed that the
statements δ [3] (F G) = 0 and δ (3) (F G) = 0 are equivalent, provided
that char(F ) ≥ 7. All the other cases the question is still open.
In general, we have very little information about the Lie derived
length of group algebras. The first and, at the same time, the more
significant results on this topic can be found in papers [27] and [29] of
A. Shalev.
Throughout this part by F G we always mean a strongly Lie solvable
group algebra.
From (1.1) it follows immediately that
(1.2) ⌈log 2(tN(G ′ ) + 1)⌉ ≤ dl L (F G) ≤ ⌈log 2(2tN(G ′ ))⌉
and so
dlL(F G) ≤ ⌈log 2(2tN(G ′ ))⌉.
Since there is no similarly valuable lower bound on dlL(F G), the computation
of its value is more difficult than of the value of dl L (F G).