{
  "id": "2008.09291",
  "version": "v1",
  "published": "2020-08-21T03:38:08.000Z",
  "updated": "2020-08-21T03:38:08.000Z",
  "title": "The non-commuting, non-generating graph of a nilpotent group",
  "authors": [
    "Peter J. Cameron",
    "Saul D. Freedman",
    "Colva M. Roney-Dougal"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "For a nilpotent group $G$, let $\\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\\Xi(G)$ has vertex set $G \\setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\\Xi^+(G)$ be the subgraph of $\\Xi(G)$ induced by its non-isolated vertices. We show that if $\\Xi(G)$ has an edge, then $\\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\\Xi(G) = \\Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\\Xi(G)$ in more detail.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-21T03:38:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F18",
      "05C25"
    ],
    "keywords": [
      "nilpotent group",
      "non-generating graph",
      "maximal subgroup normal",
      "vertex set",
      "central elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}