Counting characters above invariant characters in solvable groups

We also discuss a related question about blocks. For a prime p and a p-block B of G, we let k(B) denote the number of ordinary characters in B. It is relatively easy to show that k(B) is bounded below by k(G,D), which is the number of conjugacy classes of G that intersect the defect group D of B. In this paper we ask what can be said if equality is achieved. We show that for p-solvable groups, if k(B)=k(G,D), then B is nilpotent and thus k(B)=|Irr(D)|. In addition, we show that this result holds for many blocks of arbitrary finite groups, including all blocks of the symmetric groups. We also extend a result on fully ramified coprime actions in [5].