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Let $I$ be the augmentation ideal of $kN$, generated by the elements $n-1$. First prove its nilpotence. A nontrivial finite $p$-group has a central element $z$ of order $p$. Put $t=z-1$. In characteristic $p$, $t^p=z^p-1=0$, andInduction on $|N|$ says that the image of $I$ in this smaller group algebra is nilpotent. Thus $I^m\subseteq tkN$ for some $m$, and $I^{mp}\subseteq t^{p}kN=0$. The trivial group starts the induction. This proves nilpotence of a p-group augmentation ideal.

The quotient homomorphism kills exactly the differences of elements in each $N$-coset, soNormality of $N$ makes $I$ stable under conjugation by $G$. Moving group elements past factors from $I$ therefore gives $(IkG{)}^r=I^{r}kG$. Consequently