Finite groups with odd Sylow automizers

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.

In this new version, we modify some errors in the first version: we add Lemma 2.1 and rewrite the proof for the case $S=D_4(q)$ and $p=3$ in the proof of Lemma 3.7