{
  "id": "2205.08795",
  "version": "v1",
  "published": "2022-05-18T08:48:23.000Z",
  "updated": "2022-05-18T08:48:23.000Z",
  "title": "Degenerations and order of graphs realized by finite abelian groups",
  "authors": [
    "Rameez Raja"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2101.02059",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let G_1 and G_2 be two groups. If there exists a homomorphism \\phi from G_1 to G_2 such that \\phi(a) = b, then a group G_1 degenerates to a group G_2. In this paper, we study degeneration in graphs and show that degeneration in groups is a particular case of degeneration in graphs. We exhibit some interesting properties of degeneration in graphs. We use this concept to present a pictorial representation of graphs realized by finite abelian groups. We discus some partial orders on the set T_p_1 \\cdots p_n of all graphs realized by finite abelian p_r-groups, where each p_r, 1 \\leq r \\leq n, is a prime number. We show that each finite abelian p_r-group of rank n can be identified with saturated chains of Young diagrams in the poset T_p_1 \\cdots p_n. We present a combinatorial formula which represents the degree of a projective representation of a symmetric group. This formula determines the number of different saturated chains in T_p_1 \\cdots p_n and the number of finite abelian groups of different orders.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-18T08:48:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian groups",
      "saturated chains",
      "study degeneration",
      "partial orders",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}