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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.

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    H is a p-group, so it has a nontrivial center. H is normal, so...?2010-10-30
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    @Qiaochu: I suspect "its" refers to $G$; that is, $H\cap Z(G)$ nontrivial.2010-10-30
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    @Arturo: ah, sorry. The proof I was thinking of actually doesn't work.2010-10-30
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    There 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

4 Answers 4