On the Derived Length of Lie Solvable Group Algebras

2 CHAPTER 1
of simple groups, as special finite p-groups. The study of the unit
group of modular group algebras was started by S.A. Jennings in the
’40s, but since the solution of almost every single problem required
the elaboration of a new method, the results came very slowly. In the
more interesting cases the group of units have such a high order that
not even the computers of present day have a capacity strong enough
to deal with them. The more important results and open questions of
this area are surveyed by the article A.A. Bódi’s [9].
The investigation of the Lie properties of group algebras as a special
polinomial identity was started after the description of group algebras
satisfying a polinomial identity, but the invention of the relation between
the property of unit group and the associated Lie algebra of
group algebras led to an extended intensity of the observation in the
’80s. Under general conditions it is not easy even to decide whether
an element is a unit or not, so to determine of its inverse would be extremely
difficult, such as the computation of the group commutators.
However, the so-called Lie commutators can be calculated without the
knowledge of the elements’ inverses. Considering the results connected
to the series which are constructed with the help of Lie commutators
we can have conclusions for the corresponding series of the group
of units, for example, derived series, upper and lower central series,
etc. This method was first applied by A.A. Bódi and I.I. Khripta [6]
who obtained that the unit group of group algebras is solvable if and
only if the group algebra is Lie solvable, provided that the characteristic
p of the field is greater than three and the basic group is a
nonabelian and if it is a nontorsion group, then its p-Sylow subgroup
is infinite. Furthermore, Lie methods were used by C. Bagiński [1]
and J. Kurdics [17, 18] for the investigation of the derived length, the
nilpotency class and the Engel length of the group of units. The harmony
between the unit groups and the associated Lie algebras is also
illustrated by the following theorem of A.A. Bódi [5]: the unit group
of the modular group algebra F G of characteristic p is a bounded
Engel group if and only if F G is a bounded Engel algebra. Additional
results on the Lie structure of group algebras may be found in
[2, 7, 8, 10, 11, 13, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].