{
  "id": "1908.03347",
  "version": "v1",
  "published": "2019-08-09T07:35:43.000Z",
  "updated": "2019-08-09T07:35:43.000Z",
  "title": "Thompson-like characterization of solubility for products of finite groups",
  "authors": [
    "P. Hauck",
    "L. S. Kazarin",
    "A. Martínez-Pastor",
    "M. D. Pérez-Ramos"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, $n$-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups $G = AB$ with subgroups $A,\\ B$ such that $\\langle a, b\\rangle$ is soluble for all $a \\in A$ and $b \\in B$. In this case, the group $G$ is said to be an $\\cal S$-connected product of the subgroups $A$ and $B$ for the class $\\cal S$ of all finite soluble groups. Our main theorem states that $G = AB$ is $\\cal S$-connected if and only if $[A,B]$ is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-09T07:35:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D40",
      "20D10"
    ],
    "keywords": [
      "thompson-like characterization",
      "solubility",
      "main theorem states",
      "study finite groups",
      "thompsons theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}