Blocks of small defect

Abstract

Let $G$ be a finite solvable group, let $p$ be a prime such that $p \geq 5$ and $O_p(G)=1$, and we denote $|G|_p=p^n$, then $G$ contains a block of defect less than or equal to $\lfloor \frac {3n} 5 \rfloor$. Let $G$ be a finite solvable group and let $p^a$ be the largest power of $p$ dividing $\chi(1)$ for an irreducible character $\chi$ of $G$, we show that $|G:\bF(G)|_p \leq p^{3a}$ for $p \geq 5$.