On the derived length of the unit group of group algebras of groups with cyclic commutator subgroup

Let F be a field of characteristic 2 and G a non-abelian group with cyclic commutator subgroup of order 2(n). In this paper, the derived length of the unit group of the group algebra FG is determined for the case when the nilpotency class of G is not n + 1. Under assumption that G is of class n + 1, a necessary and sufficient condition is provided for the derived length to take its highest possible value, n + 1.