we know that if $G$ is a finite $p$-group whit $|G|\gt1$. then $Z(G)$,the center of $G$, has more than one element.for proof of this we use the $|G|=|Z(G)|+\sum[G:C(a)]$ st $a\notin Z(G)$. is the way show $Z(G)\gt1$ whit out use class equation?
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$\begingroup$ Read here: math.stackexchange.com/questions/334763/… Yet it is much easier with the class equation. $\endgroup$ – Timbuc Apr 29 '15 at 19:27
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$\begingroup$ here is a try :math.stackexchange.com/a/64978/204937 $\endgroup$ – Elaqqad Apr 29 '15 at 20:19
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