{
  "id": "1912.07279",
  "version": "v1",
  "published": "2019-12-16T10:21:16.000Z",
  "updated": "2019-12-16T10:21:16.000Z",
  "title": "Separability of Schur rings over abelian groups of odd order",
  "authors": [
    "Grigory Ryabov"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\\mathcal{K}$ is induced by a combinatorial isomorphism. A finite group $G$ is said to be separable with respect to $\\mathcal{K}$ if every $S$-ring over $G$ is separable with respect to $\\mathcal{K}$. We prove that every abelian group $G$ of order $9p$, where $p$ is a prime, is separable with respect to the class of all finite abelian groups. Modulo previously obtained results, this completes a classification of noncyclic abelian groups of odd order that are separable with respect to the class of all finite abelian groups. Also this implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-16T10:21:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05C60",
      "20B35"
    ],
    "keywords": [
      "odd order",
      "schur ring",
      "finite abelian groups",
      "separability",
      "noncyclic abelian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}