Blocks of small defect

TL;DR

This paper investigates the structure of finite solvable groups, establishing bounds on defect blocks and character degrees related to prime divisors, enhancing understanding of their representation theory.

Contribution

It provides new bounds on the defect of blocks and character degrees in finite solvable groups for primes p ≥ 5.

Findings

Existence of blocks with defect ≤ ⌊3n/5⌋ in certain groups

Bound |G:F(G)|_p ≤ p^{3a} for irreducible characters

Results applicable to groups with trivial O_p(G) and prime p ≥ 5

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 $p \geq 5$.