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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?

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    $\begingroup$ This has also been asked on MSE It looks to me as though the lecture notes are incomplete, and it might be difficult to guess what the author had in mind for a proof. $\endgroup$ – Derek Holt Jun 12 '17 at 8:52
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    $\begingroup$ I wonder whether it is possible to prove that there is an irreducible character of degree 5 (I don't see it as an immediate corollary of what's in the notes, but it might follow by a related argument). In the late 1960s, R. Brauer classified the finite groups which have an irreducible faithful complex irreducible character of degree 5, which would give only the simple groups listed as possibilities for the simple answers ( PSL(2,11) is on Brauer's list, but its order has four different prime divisors,, so can be excluded here) $\endgroup$ – Geoff Robinson Jun 12 '17 at 10:48
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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

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