{
  "id": "2108.00822",
  "version": "v1",
  "published": "2021-07-16T15:23:47.000Z",
  "updated": "2021-07-16T15:23:47.000Z",
  "title": "Extremal product-one free sequences over $C_n \\rtimes_s C_2$",
  "authors": [
    "Fabio Enrique Brochero Martínez",
    "Sávio Ribas"
  ],
  "comment": "9 pages. This is a complete answer to the inverse problem proposed in the old versions of the paper \"The $\\{1,s\\}$-weighted Davenport constant in $C_n^k$\". In those old versions, partial answers were given using the bounds of the weighted problem. This complete answer does not use any bound obtained there",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $G$ be a finite group multiplicatively written. The small Davenport constant of $G$ is the maximum positive integer ${\\sf d}(G)$ such that there exists a sequence $S$ of length ${\\sf d}(G)$ for which every subsequence of $S$ is product-one free. Let $s^2 \\equiv 1 \\pmod n$, where $s \\not\\equiv \\pm1 \\pmod n$. It has been proven that ${\\sf d}(C_n \\rtimes_s C_2) = n$ (see Lemma 6 of [Zhuang, Gao; Europ. J. Combin. 26 (2005), 1053-1059]). In this paper, we determine all sequences over $C_n \\rtimes_s C_2$ of length $n$ which are product-one free. It completes the classification of all product-one free sequences over every group of the form $C_n \\rtimes_s C_2$, including the quasidihedral groups and the modular maximal-cyclic groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-16T15:23:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P70",
      "11B50"
    ],
    "keywords": [
      "extremal product-one free sequences",
      "modular maximal-cyclic groups",
      "finite group multiplicatively written",
      "small davenport constant",
      "quasidihedral groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}