Self-normalizing Sylow subgroups

Abstract:

Using the classification of finite simple groups we prove the following statement: Let $p>3$ be a prime, $Q$ a group of automorphisms of $p$-power order of a finite group $G$, and $P$ a $Q$-invariant Sylow $p$-subgroup of $G$. If $\mathbf {C}_{\mathbf {N}_G(P)/P}(Q)$ is trivial, then $G$ is solvable. An equivalent formulation is that if $G$ has a self-normalizing Sylow $p$-subgroup with $p >3$ a prime, then $G$ is solvable. We also investigate the possibilities when $p=3$.