On the Taketa bound for normally monomial p-groups of maximal class

A longstanding problem in the representation theory of finite solvable groups, sometimes called the Taketa problem, is to find strong bounds for the derived length dl(G) in terms of the number cd(G) of irreducible character degrees of the group G. For p-groups an old result of Taketa implies that dl(G) cd(G), and while it is conjectured that the true bound is much smaller (in fact, logarithmic) for large dl(G), it turns out to be extremely difficult to improve on Taketaʹs bound at all. Here, therefore, we suggest to first study the problem for a restricted class of p-groups, namely normally monomial p-groups of maximal class. We exhibit some structural features of these groups and show that if G is such a group, then .