The following is an excerpt from section 8.3 of Représentations linéaires des groupes finis by J.-P. Serre.
$\textbf{Theorem 14.}$ Every $p$-group is nilpotent.
In view of the preceding, it suffices to show that the centre of every non-trivial $p$-group is non-trivial.
He then goes on to show that if $G$ is a $p$-group, then $p$ divides $\lvert Z(G)\rvert$, thereby proving that the centre is non-trivial.
My question is: Why does it suffice to show that the centre is non-trivial?
He says it is "in view of the preceding / vu ce qui précède", but the preceding is just a definition of various classes of finite groups (soluble/supersoluble/nilpotent/$p$-groups).
If anyone out there has read the book and understood the proof, your help would be much appreciated.
