{
  "id": "2012.09582",
  "version": "v1",
  "published": "2020-12-17T13:47:21.000Z",
  "updated": "2020-12-17T13:47:21.000Z",
  "title": "On small groups of finite Morley rank with a tight automorphism",
  "authors": [
    "Ulla Karhumäki",
    "Pınar Uğurlu"
  ],
  "comment": "30 pages",
  "categories": [
    "math.GR",
    "math.LO"
  ],
  "abstract": "We consider an infinite simple group of finite Morley rank $G$ of Pr\\\"{u}fer $2$-rank $1$ which admits a tight automorphism $\\alpha$ whose fixed-point subgroup $C_G(\\alpha)$ is pseudofinite. We prove that $C_G(\\alpha)$ contains a subgroup isomorphic to the Chevalley group ${\\rm PSL}_2(F)$, where $F$ is a pseudofinite field of characteristic $\\neq 2$. Moreover, we prove that, if a maximal split torus $T$ of ${\\rm PSL}_2(F)$ contains an involution and if $F$ is of positive characteristic, then $G \\cong {\\rm PSL}_2(K)$ for some algebraically closed field $K$ of characteristic $> 2$. These results are based on the work of the second author, where a new strategy to approach the Cherlin-Zilber Conjecture--stating that infinite simple groups of finite Morley rank are algebraic groups over algebraically closed fields--was developed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-17T13:47:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C20",
      "03C45",
      "20G99"
    ],
    "keywords": [
      "finite morley rank",
      "tight automorphism",
      "small groups",
      "infinite simple group",
      "maximal split torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}