# Finite group in which two Sylow-$p$ subgroups intersect non-trivially but their centers do not

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.

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