Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$?

Since the $\mathbb{F}_p$-vector space spanned by $n$ is exactly the annihilator of the maximal ideal, it follows that this vector space is characteristic and thus $\varphi(n)=\lambda n$ for some $\lambda \in \mathbb{F}_p^*$ and thus we have a group homomorphism: $Aut(\mathbb{F}_pG)\rightarrow \mathbb{F}_p^*$. So the question is whether this map is always trivial.


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