{
  "id": "2401.14222",
  "version": "v1",
  "published": "2024-01-25T15:02:34.000Z",
  "updated": "2024-01-25T15:02:34.000Z",
  "title": "Fixed point subgroups of a supertight automorphism",
  "authors": [
    "Ulla Karhumäki"
  ],
  "categories": [
    "math.GR",
    "math.LO"
  ],
  "abstract": "Let $G$ be an infinite simple group of finite Morley rank and $\\alpha$ a supertight automorphism of $G$ so that the fixed point subgroup $P_n:=C_G(\\alpha^n)$ is pseudofinite for all $n\\in \\mathbb{N}\\setminus\\{0\\}$. It is know (using CFSG) that the socle $S_n:={\\rm Soc}(P_n)$ is a (twisted) Chevalley group over a pseudofinite field. We prove that there is $r\\in \\mathbb{N}\\setminus\\{0\\}$ so that for each $n$ we have $[P_n:S_n] < r$ and that there is no $m \\in \\mathbb{N}\\setminus \\{0\\}$ so that for each $n$ the sizes of the Sylow $2$-subgroups of $S_n$ are bounded by $m$. We also note that in the recent identification result of $G$ under the assumption ${\\rm pr}_2(G)=1$, the use of CFSG is not needed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-25T15:02:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fixed point subgroup",
      "supertight automorphism",
      "finite morley rank",
      "infinite simple group",
      "chevalley group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}