= Nilpotence of a p-group augmentation ideal
In characteristic $p$, the <augmentation ideal> $I$ of the <group algebra> of a finite $p$-group is nilpotent. Choose a central element $z$ of order $p$ and put $t=z-1$, so $t^p=0$. The quotient by $t$ is the group algebra of the smaller quotient group. Induction gives $I^m\subseteq t kP$ for some $m$, hence $I^{mp}=0$. For a normal $p$-subgroup $P$ of $G$, conjugation stability gives $(I kG)^m=I^m kG$, proving nilpotence of the kernel of $kG\to k(G/P)$.
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