{
  "id": "2003.06624",
  "version": "v1",
  "published": "2020-03-14T13:11:47.000Z",
  "updated": "2020-03-14T13:11:47.000Z",
  "title": "On $\\BCI$-groups and $\\CI$-groups",
  "authors": [
    "Asieh Sattari",
    "Majid Arezoomand",
    "Mohammad A. Iranmanesh"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finite group and $S$ be a subset of $G.$ A bi-Cayley graph $\\BCay(G,S)$ is a simple and an undirected graph with vertex-set $G\\times\\{1,2\\}$ and edge-set $\\{\\{(g,1),(sg,2)\\}\\mid g\\in G, s\\in S\\}$. A bi-Cayley graph $\\BCay(G,S)$ is called a $\\BCI$-graph if for any bi-Cayley graph $\\BCay(G,T)$, whenever $\\BCay(G,S)\\cong\\BCay(G,T)$ we have $T=gS^\\sigma$ for some $g\\in G$ and $\\sigma\\in\\Aut(G).$ A group $G$ is called a $\\BCI$-group if every bi-Cayley graph of $G$ is a $\\BCI$-graph. In this paper, we showed that every $\\BCI$-group is a $\\CI$-group, which gives a positive answer to a conjecture proposed by Arezoomand and Taeri in \\cite{arezoomand1}. Also we proved that there is no any non-Abelian $4$-$\\BCI$-simple group. In addition all $\\BCI$-groups of order $2p$, $p$ a prime, are characterized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-14T13:11:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bi-cayley graph",
      "finite group",
      "simple group",
      "undirected graph",
      "vertex-set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}