{
  "id": "2303.09940",
  "version": "v1",
  "published": "2023-03-17T12:54:31.000Z",
  "updated": "2023-03-17T12:54:31.000Z",
  "title": "The socle of the group algebra of a finite $p$-group",
  "authors": [
    "David J. Benson"
  ],
  "comment": "3 pages",
  "categories": [
    "math.RT",
    "math.GR",
    "math.RA"
  ],
  "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 $(k^\\times)^{p-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-17T12:54:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C20"
    ],
    "keywords": [
      "group algebra",
      "algebra automorphism",
      "linear action",
      "dimension subquotients",
      "scalar multiplication"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}