Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 5 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.