# Concrete Example of Sylow Subgroups which intersect nontrivially.

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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Have you tried some examples of groups you're familiar with to see if their Sylow subgroups have this property? –  Qiaochu Yuan Oct 2 '12 at 1:19
Up until this point in the course, the only concrete examples we have seen of groups are cyclic groups; and I can't think of an example from drawing out some of the smaller ones on paper. –  Steve Oct 2 '12 at 1:53
You know what a Sylow subgroup is but the only concrete examples of groups you know are cyclic groups?! Are you familiar with the symmetric groups, at least? –  Qiaochu Yuan Oct 2 '12 at 2:01
We are talking about those this week simultaneously, so once I can picture them better I can look there for examples as well. I guess I was more looking for a method to find such a group, if there is one, like if there are any general properties that would help you notice if three Sylow subgroups will end up behaving this way. I can always try to run through a list of different groups I come across, but that doesn't seem like it would be as enlightening if there's a better way. –  Steve Oct 2 '12 at 2:17
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. –  Qiaochu Yuan Oct 2 '12 at 2:23

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.