{
  "id": "2201.13244",
  "version": "v1",
  "published": "2022-01-28T07:35:10.000Z",
  "updated": "2022-01-28T07:35:10.000Z",
  "title": "A generalization of a question asked by B. H. Neumann",
  "authors": [
    "Andrea Lucchini"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $w \\in F_2$ be a word and let $m$ and $n$ be two positive integers. We say that a finite group $G$ has the $w_{m,n}$-property if however a set $M$ of $m$ elements and a set $N$ of $n$ elements of the group is chosen, there exist at least one element of $x \\in M$ and at least one element of $y \\in M$ such that $w(x,y)=1.$ Assume that there exists a constant $\\gamma < 1$ such that whenever $w$ is not an identity in a finite group $X$, then the probability that $w(x_1,x_2)=1$ in $X$ is at most $\\gamma.$ If $m\\leq n$ and $G$ satisfies the $w_{m,n}$-property, then either $w$ is an identity in $G$ or $|G|$ is bounded in terms of $\\gamma, m$ and $n$. We apply this result to the 2-Engel word.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-28T07:35:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalization",
      "finite group",
      "positive integers",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}