Nontrivial center of a finite p-group
= Nontrivial center of a finite p-group
Every nontrivial <finite p-group> has a nontrivial <center of a group>[center]. The conjugation action partitions the group into conjugacy classes whose noncentral sizes are positive powers of $p$; the class equation then implies $|Z(G)|\equiv|G|\equiv0\pmod p$.