{
  "id": "1803.10179",
  "version": "v3",
  "published": "2018-03-27T16:53:45.000Z",
  "updated": "2018-08-23T10:03:24.000Z",
  "title": "Integrals of groups",
  "authors": [
    "João Araújo",
    "Peter J. Cameron",
    "Carlo Casolo",
    "Francesco Matucci"
  ],
  "comment": "31 pages, no figures; new co-author and new title; the previous posting has been split in half, with the second part to be expanded and resubmitted separately",
  "categories": [
    "math.GR"
  ],
  "abstract": "An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: (1) If a finite group has an integral, then it has a finite integral. (2) A precise characterization of the set of natural numbers $n$ for which every group of order $n$ is integrable: these are the cubefree numbers $n$ which do not have prime divisors $p$ and $q$ with $q\\mid p-1$. (3) An abelian group of order $n$ has an integral of order at most $n^{1+o(1)}$, but may fail to have an integral of order bounded by $cn$ for constant $c$. (4) A finite group can be integrated $n$ times (in the class of finite groups) if and only if it is the central product of an abelian group and a perfect group. There are many other results on such topics as centreless groups, groups with composition length $2$, and infinite groups. We also include a number of open problems.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2018-08-23T10:03:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D99",
      "20D25"
    ],
    "keywords": [
      "abelian group",
      "paper discusses integrals",
      "cubefree numbers",
      "main results",
      "commutator subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}