{
  "id": "2409.03399",
  "version": "v1",
  "published": "2024-09-05T10:30:24.000Z",
  "updated": "2024-09-05T10:30:24.000Z",
  "title": "On Heisenberg groups",
  "authors": [
    "Florian L. Deloup"
  ],
  "comment": "10 pages. Preliminary",
  "categories": [
    "math.GR"
  ],
  "abstract": "It is known that an abelian group $A$ and a $2$-cocycle $c:A \\times A \\to C$ yield a group ${\\mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In this note, we prove that under mild conditions, any class~$2$ nilpotent group $G$ is equivalent as an extension of $G/[G,G]$ to a Heisenberg group ${\\mathscr{H}}(G/[G,G], [G,G], c')$ whose $2$-cocycle $c'$ is bimultiplicative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-05T10:30:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "20F18"
    ],
    "keywords": [
      "heisenberg group",
      "nilpotent group",
      "mild conditions",
      "abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}