If $G$ is a finite $p$-group with a nontrivial normal subgroup $H$, then the intersection of $H$ and the center of $G$ is not trivial.
A normal subgroup intersects the center of the $p$-group nontrivially
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group-theory
finite-groups
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0H is a p-group, so it has a nontrivial center. H is normal, so...? – 2010-10-30
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0@Qiaochu: I suspect "its" refers to $G$; that is, $H\cap Z(G)$ nontrivial. – 2010-10-30
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0@Arturo: ah, sorry. The proof I was thinking of actually doesn't work. – 2010-10-30
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1There is a nice generalisation of this result. If $G$ is a nilpotent group and $1\neq H\unlhd G$, then $H\cap Z(G)\neq 1$. Since all $p$-groups are nilpotent, your result could be seen as a corollary of this (if you want a different way of looking at things that is). – 2013-05-17