Character degrees and derived length in p-groups

Abstract

There are a number of theorems which bound d.l.(G), the derived length of a group G, in terms of the size of the set c.d.(G) of irreducible character degrees of G assuming that G is in some particular class of solvable groups ([1], [3], [4], [7]). For instance, Gluck [4] shows that d.l.(G)≤2 |c.d.(G)| for any solvable group, whereas Berger [1] shows that d.l.(G)≤|c.d.(G)| if G has odd order. One of the oldest (and smallest) such bounds is a theorem of Taketa [7] which says that d.l.(G)≤|c.d.(G)| if G is an M-group. Most of the existing theorems are an attempt to extend Taketa's bound to all solvable groups. However, it is not even known for M-groups whether or not this is the best possible bound. This suggests that given a class of solvable groups one might try to find the maximum derived length of a group with n character degrees (i.e. the best possible bound).