In [Lectures by Dan Bump on Modular representation theory](http://web.stanford.edu/~tonyfeng/mod_rep_theory.pdf),
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?



Group of order $5p^aq^b$
Note that Bump has his own notes in his webpage: <http://sporadic.stanford.edu/modrep>


While the contents are in different order with that of Feng's notes, he mentioned in Section~6.3 that this is a result of Brauer.


I found that it is in this paper of Brauer: <http://www.ams.org/journals/bull/1968-74-05/S0002-9904-1968-12073-7/home.html>


