In Lectures by Dan Bump on Modular representation theory, Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-Sylow $R$ is equal to its own centralizer.

He states Corollary 13.15 (with no proof), namely when $r=5$, it follows that $G$ is isomorphic to $A_5$, $A_6$, or $\mathrm{SO}_5(\mathbb{F}_3)$.

How can one deduce the corollary? Is there a paper/book that I can refer to?