If $ G$ is an almost simple finite group, then $G$ has no nontrivial normal abelian Sylow subgroup

Question

Why the almost simple finite groups has no nontrivial normal abelian Sylow subgroups? Any illustrations or recommended books to understand this idea?

Answer 1

Let $S\lhd G$ be the non-abelian simple group, and let $P\lhd G$ be a normal Sylow subgroup. Note that $S\cap P\lhd S$, so $S\cap P=1$. Thus $P$ centralizes $S$, which means it acts trivially on $S$. Since $G$ is almost simple, $P$ is trivial.

Note that $P$ being abelian doesn't matter here. What's important is that $P$ cannot contain a non-abelian simple subgroup (so that $S\cap P\neq S$). Thus, the same argument shows $G$ contains no normal solvable subgroup.