{
  "id": "2005.03634",
  "version": "v1",
  "published": "2020-05-07T17:39:31.000Z",
  "updated": "2020-05-07T17:39:31.000Z",
  "title": "Word problems for finite nilpotent groups",
  "authors": [
    "Rachel D. Camina",
    "Ainhoa Iniquez",
    "Anitha Thillaisundaram"
  ],
  "comment": "9 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $w$ be a word in $k$ variables. For a finite nilpotent group $G$, a conjecture of Amit states that $N_w(1) \\ge |G|^{k-1}$, where $N_w(1)$ is the number of $k$-tuples $(g_1,...,g_k)\\in G^{(k)}$ such that $w(g_1,...,g_k)=1$. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, and prove that $N_w(g) \\ge |G|^{k-2}$, where $g$ is a $w$-value in $G$, for finite groups $G$ of odd order and nilpotency class 2. If $w$ is a word in two variables, we further show that $N_w(g) \\ge |G|$, where $g$ is a $w$-value in $G$ for finite groups $G$ of nilpotency class 2. In addition, for $p$ a prime, we show that finite $p$-groups $G$, with two distinct irreducible complex character degrees, satisfy the generalized Amit conjecture for words $w_k =[x_1,y_1]...[x_k,y_k]$ with $k$ a natural number; that is, for $g$ a $w_k$-value in $G$ we have $N_{w_k}(g) \\ge |G|^{2k-1}$. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-07T17:39:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F10",
      "20D15"
    ],
    "keywords": [
      "finite nilpotent group",
      "word problems",
      "nilpotency class",
      "finite group",
      "distinct irreducible complex character degrees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}