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How would I go about finding an example of a finite group G having 3 p-Sylow subgroups of the same order (say P, Q and R) such that P and Q intersect trivially but Q and R do not?

I understand how to see that sometimes p-Sylow subgroups must nontrivially intersect, but am having trouble figuring out how to come up with concrete examples.

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    Running through a list of groups can be extremely enlightening. Anyway, the point is that you know that a group acts transitively on its Sylow $p$-subgroups but it isn't guaranteed to act $2$-transitively, so you expect that if you just pick a random nonabelian group it will probably have this behavior and so you should just go through the ones you know and see what happens.2012-10-02

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Useless mathematician answer: Take $Q$ and $R$ to be the same group and take any $P$ that intersects trivially with them.

But, more seriously, here is an example from $S_5$. Let $Q$ and $R$ be the $2$-Sylow subgroups $Q=\langle (1,2),(1,3)(2,4)\rangle$ and $R=\langle (4,5),(1,4)(2,5)\rangle$. These intersect nontrivially: $Q \cap R = \langle (1,2) \rangle$. Now, let $P=\langle (3,4),(1,4)(3,5)\rangle$ and $P \cap Q = 1$ as required.

Edit: Just for fun I found the smallest group which has a counterexample of the type you mention. It's $S_3 \times S_3$. If we represent the group by $S_3 \times S_3=\langle (1,2,3),(1,2),(4,5,6),(4,5)\rangle$, we've got $P=\langle (1,2),(4,6) \rangle$, $Q=\langle (2,3),(5,6) \rangle$, and $R = \langle(2,3),(4,5)\rangle$. This is probably a better example anyway.