On the Derived Length of Lie Solvable Group Algebras

Chapter 4
Group algebras of maximal Lie
derived length in characteristic
two
According to Proposition 2.1.1, if F G is a Lie solvable group algebra
then
dlL(F G) ≤ ⌈log 2(2tN(G ′ ))⌉.
For characteristic two, if G is a nilpotent group with commutator
subgroup of order 2 n , we obtain here the description of the group
algebras F G which have the highest possible value of dlL(F G), namely,
⌈log 2(2tN(G ′ ))⌉ = n + 1.
4.1 Preliminaries
Let G be a group with commutator subgroup G ′ = 〈x | x2n = 1〉.
First of all, note that G is a nilpotent group. Indeed, let m ≥ 1
and γm(G) = 〈x2k〉. Then the subgroup γm+1(G) is generated by the
commutators (x2k, g), where g ∈ G. Clearly,
(x 2k
, g) = x −2k
(x 2k
) g = x −2k
x l2k
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= x 2k (−1+l)