The socle of the group algebra of a finite $p$-group

Abstract

Let $G$ be a finite $p$-group, and $\alpha$ an automorphism of the group algebra ${\mathbb F}_pG$. Then $\alpha$ fixes the socle of ${\mathbb F}_pG$ pointwise. More generally, if $k$ is a field of characteristic $p$, and $\alpha$ is a $k$-algebra automorphism of $kG$, then $\alpha$ induces a linear action on the dimension subquotients of the group, and the action on the socle is scalar multiplication by the $(p-1)$st power of the product of the determinants of this action. The scalar is thus an element of $(k^\times)^{p-1}$.