{
  "id": "2407.14511",
  "version": "v1",
  "published": "2024-06-18T17:38:20.000Z",
  "updated": "2024-06-18T17:38:20.000Z",
  "title": "Counting subgroups of the groups ${\\Bbb Z}_{n_1} \\times \\cdots \\times {\\Bbb Z}_{n_k}$: a survey",
  "authors": [
    "László Tóth"
  ],
  "comment": "book chapter. arXiv admin note: text overlap with arXiv:1611.03302",
  "journal": "New Frontiers in Number Theory and Applications, Trends in Mathematics, Birkh\\\"auser, Cham, 2024, Gu\\`ardia, J., Minculete, N., Savin, D., Vela, M.,Zekhnini, A. (eds), pp. 385-409",
  "doi": "10.1007/978-3-031-51959-8_18",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "We present a survey of exact and asymptotic formulas on the number of cyclic subgroups and total number of subgroups of the groups ${\\Bbb Z}_{n_1} \\times \\cdots \\times {\\Bbb Z}_{n_k}$, where $k\\ge 2$ and $n_1,\\ldots,n_k$ are arbitrary positive integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-18T17:38:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20K01",
      "20K27",
      "05A15",
      "11A25"
    ],
    "keywords": [
      "counting subgroups",
      "arbitrary positive integers",
      "total number",
      "asymptotic formulas"
    ],
    "tags": [
      "book chapter",
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}