Blocks of defect of p-solvable groups

Abstract

Let $p$ be a prime such that $p\ge 5$. Let $G$ be a finite $p$-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 $|G:F(G){|}_p\le p^{5.5a}$. Let $G$ be a finite $p$-solvable group with trivial maximal normal solvable subgroup and we denote $|G{|}_p=p^n$, then $G$ contains a block of defect less than or equal to $\lfloor \frac{2n}{3}\rfloor$.

arXiv admin note: substantial text overlap with arXiv:1208.4022