Blocks of defect of p-solvable groups

TL;DR

This paper investigates the structure of p-solvable groups, establishing bounds on the index of the Fitting subgroup related to character degrees and identifying blocks with bounded defect, advancing understanding of group representation theory.

Contribution

It provides new bounds on the p-part of the index of the Fitting subgroup and identifies blocks with bounded defect in p-solvable groups with trivial solvable radical.

Findings

|G:F(G)|_p ext{ is bounded by } p^{5.5a}

Existence of blocks with defect extless= loor{rac{2n}{3}}

Results apply to groups with trivial maximal normal solvable subgroup

Abstract

Let $p$ be a prime such that $p \geq 5$. 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 $|G:F(G)|_p \leq p^{5.5a}$. 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 $\lfloor \frac {2n} {3} \rfloor$.