{
  "id": "1808.08708",
  "version": "v1",
  "published": "2018-08-27T07:11:38.000Z",
  "updated": "2018-08-27T07:11:38.000Z",
  "title": "Cardinality of product of sets in torsion-free groups and applications in group algebras",
  "authors": [
    "Alireza Abdollahi",
    "Fatemeh Jafari"
  ],
  "categories": [
    "math.GR",
    "math.NT",
    "math.RA"
  ],
  "abstract": "Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and $B$ of $G$ there exists $x\\in G$ which can be uniquely expressed as a product of an element of $A$ and an element of $B$. We prove that, if $C$ is a finite subset of $G$ containing the identity element such that $\\langle C\\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\\geq 7$, $|BC|\\geq |B| + |C| + 2$. Also, we prove that if $C$ is a finite subset containing the identity element of a torsion-free group $G$ such that $|C| = 3$ and $\\langle C\\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\\geq 7$, $|BC|\\geq |B| + 5$. Moreover, if $\\langle C\\rangle$ is not isomorphic to the Klein bottle group, i.e., the group with the presentation $\\langle x, y \\;|\\; xyx = y\\rangle$, then for all subsets $B$ of G with $|B|\\geq 5$, $|BC|\\geq |B| + 5$. The support of an element $\\alpha =\\sum_{x\\in G} \\alpha_x x$ a group algebra $F[G]$ ($F$ is any field), denoted by $supp(\\alpha)$, is the set $\\{x\\in G \\;|\\; \\alpha_x \\neq 0\\}$. By the latter result, we prove that if $\\alpha \\beta = 0$ for some non-zero $\\alpha,\\beta \\in F[G]$ such that $|supp(\\alpha)| = 3$, then $|supp(\\beta)|\\geq 12$. Also, we prove that if $\\alpha \\beta= 1$ for some $\\alpha \\beta \\in F[G]$ such that $|supp(\\alpha)| = 3$, then |supp(\\alpha)|\\geq 10$. These results improve a part of results in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667-693] and Dykema et al. [Exp. Math., 24 (2015), 326-338] to arbitrary fields, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-27T07:11:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D60",
      "20C05",
      "11P70",
      "16S34"
    ],
    "keywords": [
      "group algebra",
      "torsion-free group",
      "finite subset",
      "identity element",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}