Let G be a finite group. It is well-known that the elements of the commutator subgroup must be products of commutators, but need not themselves be commutators. A natural question is to determine the minimal integer, λ( G), such that each element of the commutator subgroup may be represented as a product of λ(G) commutators. An analysis of a known character identity allows us to improve the existing lower bounds for |G| in terms of λ(G). The techniques we develop also address the related following question. Suppose we have a conjugacy class C of a finite group G such that ⟨C⟩ = G = G'. One may ask for the minimal integer cn(C) such that each element of G may be expressed as a product of cn( C) elements of the conjugacy class. Again, we improve the known upper bounds, this time for cn(C).

Our second focus is the relation between the derived length of a finite solvable group and the cardinality of the set of character degrees in the same group. Over the past few decades, this topic has been explored by Isaacs, Gluck, Slattery, and most recently, by Thomas Keller. There is a standing conjecture that universal constants C₁ and C₂ exist such that for any finite solvable group G, dlG≤C 1logcdG +C2. Indeed, Thomas Keller has reduced the conjecture to the case of p-groups, and proceeded to attack this case by a study of normally monomial p-groups of maximal class. We extend and refine his methods to a broader class of groups, those for which each irreducible character may be induced from a single normal series. We also examine the special properties held by these groups, said to be normally serially monomial.