Why do Sylow $p$-groups in finite simple group have trivial intersection?

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.

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