On the Derived Length of Lie Solvable Group Algebras

A POSSIBILITY OF APPLICATION 57
Indeed, apart from the cases i ∈ {17, 26}, G ′ i is cyclic. If either
i ∈ {2, 3} and m = 4 or i ∈ {1, 4, 5, 9, 10} then G ′ i = C2 and
the statement follows from e. g. Theorem 3.3.1. Furthermore, if
i ∈ {15, 16, 18, 20, 24, 25} and m > 5 then Gi has a normal subgroup
with commutator subgroup of order two. Using Theorem 5.1.2 it follows
that dlL(F Gi) ≤ 4. To prove the converse inequality we can use
Theorem 5.1.6. In all the other cases either G ′ i is of order 4 or Gi
has an abelian subgroup of index two. Then Theorem 5.1.6 guaranteers
the statement. Finally, G ′ 17 ∼ = G ′ 26 ∼ = C2 × C2 and we can apply
Proposition 2.1.2 to compute the derived length.