{
  "id": "2108.02021",
  "version": "v1",
  "published": "2021-08-04T12:45:01.000Z",
  "updated": "2021-08-04T12:45:01.000Z",
  "title": "Probabilistically nilpotent groups of class two",
  "authors": [
    "Sean Eberhard",
    "Pavel Shumyatsky"
  ],
  "comment": "20 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \\in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \\geq \\epsilon > 0$, $G$ has a class-4 nilpotent normal subgroup $H$ such that $[G : H] $ and $|\\gamma_4(H)|$ are both bounded in terms of $\\epsilon$. We also show that if $G$ is an infinite group whose commutators have boundedly many conjugates, or indeed if $G$ satisfies a certain more general commutator covering condition, then $G$ is finite-by-class-3-nilpotent-by-finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-04T12:45:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F24",
      "20F18",
      "20F69",
      "20P05"
    ],
    "keywords": [
      "probabilistically nilpotent groups",
      "general commutator covering condition",
      "nilpotent normal subgroup",
      "infinite group",
      "conjugates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}