{
  "id": "2309.12791",
  "version": "v1",
  "published": "2023-09-22T11:04:15.000Z",
  "updated": "2023-09-22T11:04:15.000Z",
  "title": "Extensible endomorphisms of compact groups",
  "authors": [
    "Alexandru Chirvasitu"
  ],
  "comment": "14 pages + references",
  "categories": [
    "math.GR",
    "math.CT",
    "math.RT"
  ],
  "abstract": "We show that the endomorphisms of a compact group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due to Schupp and Pettet on discrete groups (plain or finite). A somewhat more surprising result is that if $\\mathbb{A}$ is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include $-\\mathrm{id}$ (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-22T11:04:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22C05",
      "22D35",
      "22D45",
      "20K30",
      "20J06",
      "20K27",
      "20C25",
      "20K40",
      "20K35",
      "20K45",
      "20G41"
    ],
    "keywords": [
      "compact group",
      "extensible endomorphisms",
      "compact connected groups",
      "inner automorphisms",
      "compact overgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}