{
  "id": "2206.14549",
  "version": "v1",
  "published": "2022-06-29T11:54:29.000Z",
  "updated": "2022-06-29T11:54:29.000Z",
  "title": "Algebraic Groups over Finite Fields: Connections Between Subgroups and Isogenies",
  "authors": [
    "Davide Sclosa"
  ],
  "comment": "10 pages",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "Let G be a linear algebraic group defined over a finite field F_q. We present several connections between the isogenies of G and the finite groups of rational points G(F_q^n). We show that an isogeny from G' to G over F_q gives rise to a subgroup of fixed index in G(F_q^n) for infinitely many n. Conversely, we show that if G is reductive the existence of a subgroup of fixed index k for infinitely many n implies the existence of an isogeny of order k. In particular, we show that every infinite sequence of subgroups is controlled by a finite number of isogenies. This result applies to classical groups GLm, SLm, SOm, SUm, Sp2m and can be extended to non-reductive groups if k is prime to the characteristic. As a special case, we see that if G is simply connected the minimal indexes of proper subgroups of G(F_q^n) diverge to infinity. Similar results are investigated regarding the sequence G(F_p) by varying the characteristic p.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-29T11:54:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G40"
    ],
    "keywords": [
      "finite field",
      "connections",
      "linear algebraic group",
      "fixed index",
      "rational points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}