On the Derived Length of Lie Solvable Group Algebras

Contents
1 Introduction 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Associated Lie algebra of group algebras . . . . . . . . 5
1.2.1 Series in the associated Lie algebra . . . . . . . 6
1.2.2 Lie derived lengths of group algebras . . . . . . 8
1.2.3 Lie nilpotency indices of group algebras . . . . . 10
2 An extension of a result of A. Shalev 11
2.1 Preliminary results . . . . . . . . . . . . . . . . . . . . 11
2.2 The generalized result . . . . . . . . . . . . . . . . . . 12
3 Lie derived lengths of group algebras of groups with
cyclic commutator subgroup 19
3.1 The basic group is nilpotent . . . . . . . . . . . . . . . 19
3.2 The basic group is not nilpotent . . . . . . . . . . . . . 23
3.3 Summarized result . . . . . . . . . . . . . . . . . . . . 36
4 Group algebras of maximal Lie derived length in characteristic
two 39
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 The description and some consequences . . . . . . . . . 45
5 Group algebras of Lie derived length three 49
5.1 Preliminaries and the description . . . . . . . . . . . . 49
5.2 Group algebras of 2-groups of order 2 m and exponent
2 m−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
VII