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

Abstract

Let G be a finite p-group, and α an automorphism of the group algebra FpG. Then α fixes the socle of FpG pointwise. More generally, if k is a field of characteristic p, and α is a k-algebra automorphism of kG, then α 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×)p−1.