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In a non-nilpotent group, some Sylow subgroup is not normal. Suppose $P_1$ and $P_2$ are two Sylow-$p$ subgroups. They may intersect, and I think, this intersection of Sylow-$p$ subgroups has been studied by many people.

If they intersect, then it is natural to consider where they really intersect? If we consider the upper central series of Sylow-$p$ subgroup, then the first term will be the center. I wonder, do two Sylow-$p$ subgroups always intersect in their centers? To be more precise,

Question: Is there an example of a finite group $G$ in which two Sylow-$p$ subgroups intersect non-trivially but their centers intersects trivially.

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    $\begingroup$ Yes. Try $G=S_5$ and $p=2$. $\endgroup$ – Derek Holt Oct 12 '15 at 11:59
  • $\begingroup$ OK. I will try this example. $\endgroup$ – Groups Oct 12 '15 at 12:32

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