{
  "id": "2404.03895",
  "version": "v1",
  "published": "2024-04-05T05:19:54.000Z",
  "updated": "2024-04-05T05:19:54.000Z",
  "title": "Genus and crosscap of Normal subgroup based power graphs of finite groups",
  "authors": [
    "Parveen",
    "Manisha",
    "Jitender Kumar"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let $H$ be a normal subgroup of a group $G$. The normal subgroup based power graph $\\Gamma_H(G)$ of $G$ is the simple undirected graph with vertex set $V(\\Gamma_H(G))= (G\\setminus H)\\cup \\{e\\}$ and two distinct vertices $a$ and $b$ are adjacent if either $aH = b^m H$ or $bH=a^nH$ for some $m,n \\in \\mathbb{N}$. In this paper, we continue the study of normal subgroup based power graph and characterize all the pairs $(G,H)$, where $H$ is a non-trivial normal subgroup of $G$, such that the genus of $\\Gamma_H(G)$ is at most $2$. Moreover, we determine all the subgroups $H$ and the quotient groups $\\frac{G}{H}$ such that the cross-cap of $\\Gamma_H(G)$ is at most three.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-05T05:19:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "power graph",
      "finite groups",
      "non-trivial normal subgroup",
      "simple undirected graph",
      "vertex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}