On the Derived Length of Lie Solvable Group Algebras

Chapter 1
Introduction
1.1 Preliminaries
At the beginning of the last century an exciting algebraic construction
consisting of a group and a field, appeared in the works of
G. Frobenius which he used for the observation of the representations
of finite groups. This is the construction that was named group algebra
in the ’30s by E. Noether. After this, through decades the group
algebras were not so much observed for themselves but for the possibilities
of their applications in representation theory and algebraic
topology, etc. For example, W. Magnus proved his famous theorem
on the lower central series of free groups at that time with the aid
of group algebras, which theorem was demonstrated using pure group
theoretical methods only after a long time. But in the beginning of the
’60s the stress was laid upon the group algebras of infinite groups as a
result of I. Kaplansky’s problems in ring theory. The characterization
theorems born in this period have led to newer directions in research.
Several survey articles were published which brought the formation of
this new branch of science. The study of the ring theoretical properties
of group algebras is now at the most advanced level, but significant
results are known in their unit groups as well. The unit group of group
algebras aroused the researchers’ interest for the first time because of
their topological applications, and then again, after the description
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