On the Derived Length of Lie Solvable Group Algebras

1.1 PRELIMINARIES 3
Our goal in this thesis is the investigation of the derived length
and the upper Lie nilpotency index of group algebras. Before we
present the new results we give a short survey of the basic concepts
and notations. For details we refer the reader to the book of A.A.
Bódi [3].
Let G be a group and let F be a field. Denote by F G all the
formal sums
g∈G αgg, where only finitely many coefficients αg ∈ F
are nonzero. Clearly, two formal sums are equal if and only if all
corresponding coefficients of group elements are equal. Let us define
the sum of x =
g∈G αgg ∈ F G and y =
g∈G βgg ∈ F G as
x + y =
(αg + βg)g
g∈G
and the product of β ∈ F and x as
β · x = x · β =
g∈G (βαg)g.
Then F G can be considered as a vector space over F and the elements
of G form an F -basis for F G. The multiplication of formal sums are
defined as follows:
xy =
g.
g∈G
h∈G
αhβ h −1 g
With these operations F G is an algebra over the field F which is called
group algebra (of the group G over the field F ).
In the special case when F is a field of characteristic char(F ) = p
and G contains an element of order p, F G is called modular group
algebra.
Let x =
g∈G αgg be a nonzero element of the group algebra F G.
The subset {g ∈ G | αg = 0} of the group G is said to be the support
of x.
The function ε mapping F G into F given by
ε αgg =
αg.
g∈G
g∈G