Finite groups with odd Sylow automizers

TL;DR

This paper characterizes the composition factors of finite groups with odd Sylow automizers and proves several important conjectures, including McKay and Alperin weight conjectures, for these groups.

Contribution

It provides a classification of nonabelian composition factors and verifies key conjectures for groups with odd Sylow automizers.

Findings

Characterization of nonabelian composition factors

Proof of McKay conjecture for these groups

Proof of Alperin weight and Alperin-McKay conjectures

Abstract

Let $p$ be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd $p$-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture and the Alperin-McKay conjecture hold for such group.