= Central intersection property of normal subgroups of finite p-groups
{title2=$1\ne N\trianglelefteq G\ \Longrightarrow\ N\cap Z(G)\ne\{1\}$}
A nontrivial <normal subgroup> $N$ of a <finite p-group> meets the <center of a group> nontrivially. The <conjugation action> of $G$ on $N$ has singleton orbits exactly at $N\cap Z(G)$. The <orbit-stabilizer theorem> makes every other orbit size divisible by $p$, so $|N\cap Z(G)|\equiv|N|\equiv0\pmod p$. Since the intersection contains the identity, its size is at least $p$.
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