{
  "id": "2310.06516",
  "version": "v1",
  "published": "2023-10-10T11:00:22.000Z",
  "updated": "2023-10-10T11:00:22.000Z",
  "title": "On the order sequence of a group",
  "authors": [
    "Peter J. Cameron",
    "Hiranya Kishore Dey"
  ],
  "comment": "22 pages, Comments are most welcome",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. Abelian groups are uniquely determined by their order sequences, and the poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-10T11:00:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15",
      "20D60",
      "20E22",
      "05E16"
    ],
    "keywords": [
      "order sequence",
      "abelian groups",
      "finite group",
      "natural partial order",
      "non-decreasing order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}