Group of order $5p^aq^b$

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According to http://web.stanford.edu/~tonyfeng/mod_rep_theory.pdf. We have in Theorem 13.14. If $G$ is a non-abelian simple group and $|G | = p^a q^b r$ for distinct primes $p$,$q$, and $r$ , then if $R$ is an $r$ -Sylow we have $R = C(R)$. I want to ask how did we conclude Corollary 13.15. Is there a paper/book that I can refer to?