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