{
  "id": "2109.06993",
  "version": "v1",
  "published": "2021-09-14T22:14:27.000Z",
  "updated": "2021-09-14T22:14:27.000Z",
  "title": "On non-normal subgroup perfect codes",
  "authors": [
    "Angelot Behajaina",
    "Roghayeh Maleki",
    "Andriaherimanana Sarobidy Razafimahatratra"
  ],
  "comment": "6 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $X = (V,E)$ be a graph. A subset $C \\subseteq V(X)$ is a \\emph{perfect code} of $X$ if $C$ is a coclique of $X$ with the property that any vertex in $V(X)\\setminus C$ is adjacent to exactly one vertex in $C$. Given a finite group $G$ with identity element $e$ and $H\\leq G$, $H$ is a \\emph{subgroup perfect code} of $G$ if there exists an inverse-closed subset $S \\subseteq G\\setminus \\{e\\}$ such that $H$ is a perfect code of the Cayley graph $\\operatorname{Cay}(G,S)$ of $G$ with connection set $S$. In this short note, we give an infinite family of finite groups $G$ admitting a non-normal subgroup perfect code $H$ such that there exists $ g\\in G$ with $g^2\\in H$ but $(gh)^2 \\neq e$, for all $h \\in H$; thus, answering a question raised by Wang, Xia, and Zhou in [Perfect sets in Cayley graphs. {\\it arXiv preprint} arXiv:2006.05100, 2020].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-14T22:14:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C69",
      "94B25"
    ],
    "keywords": [
      "non-normal subgroup perfect code",
      "cayley graph",
      "finite group",
      "identity element",
      "short note"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}