Do $p$-Sylow and $q$-Sylow subgroups commute iff both are unique and thus normal? I know that one direction is true: namely that if the $p$-Sylow subgroup and the $q$-Sylow subgroup are normal in the group, that they commute.

My question: Is the other direction also true?

  • $\begingroup$ It is enough that only one of them is normal for them to commute. $\endgroup$ – Timbuc Oct 23 '14 at 23:07
  • $\begingroup$ I am especially interested in a specific case. Really I want to show that given a group $G$ of order $p(p+1)$ with $p+1$ Sylow $p$-subgroups, any given Sylow $p$-subgroup acts nontrivially on the set of non-id elements with order not $p$. If what you're saying is true then I don't think this is true. $\endgroup$ – Johann Linus Oct 23 '14 at 23:09
  • $\begingroup$ It is, @Johann...but let's make sure we both have the same definitions: two subgroups $\;H\,,\,K\;$ of a group $\;G\;$ are said to commute if $$\;HK=KH\iff \forall\,h\in H\,,\,\,jk\in K\;\;\exists\,h'\in H\,,\,\,k\in K\;\;s.t.\;\;hk=k'h'\;$$ The above is what I meant with commuting subgroups. $\endgroup$ – Timbuc Oct 23 '14 at 23:10
  • $\begingroup$ @Timbuc I believe you intended to say more? :) $\endgroup$ – Johann Linus Oct 23 '14 at 23:11
  • $\begingroup$ In fact, we know that if $\;H\,,\,K\le G\;$ ,then $\;HK\;$ is also a subgroup of $\;G\;\iff\; HK=KH\;$ $\endgroup$ – Timbuc Oct 23 '14 at 23:12

Counterexample: $G=S_3$, $p=2$ and $q=3$.


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