I'm looking for an argument for the fact that two Sylow $p$-subgroups in a finite simple group have trivial intersection. In the case that the order of the Sylow subgroups is the prime $p$ it is easy, but I don't see how I can prove it in the general case of a prime power.

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    $\begingroup$ You won't be able to prove it because it is not true in general. $\endgroup$ – Derek Holt Jan 9 '15 at 17:34
  • $\begingroup$ @DerekHolt yes, it is. He is talking about simple groups. $\endgroup$ – Jonathan Hebert Jan 9 '15 at 17:38
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    $\begingroup$ $\langle (1,2,3),(4,5,6) \rangle$ and $\langle (1,2,3),(5,6,7) \rangle$ are two Sylow $3$-subgroups of $A_7$ with non-trivial intersection. $\endgroup$ – Derek Holt Jan 9 '15 at 17:41
  • $\begingroup$ I was mistaken, I conflated "intersection of conjugate subgroups" with "intersection of all conjugate subgroups." $\endgroup$ – Jonathan Hebert Jan 9 '15 at 17:52
  • $\begingroup$ @DerekHolt and JonathanHebert Thank you! $\endgroup$ – Geometer61 Jan 10 '15 at 11:20

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