{
  "id": "1709.03937",
  "version": "v1",
  "published": "2017-09-12T16:21:38.000Z",
  "updated": "2017-09-12T16:21:38.000Z",
  "title": "On separability of Schur rings over abelian p-groups",
  "authors": [
    "Grigory Ryabov"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "An $S$-ring (Schur ring) is called separable with respect to a class of $S$-rings $\\mathcal{K}$ if it is determined up to isomorphism in $\\mathcal{K}$ only by the tensor of its structure constants. An abelian group is said to be separable if every $S$-ring over this group is separable with respect to the class of $S$-rings over abelian groups. Let $C_n$ be a cyclic group of order $n$ and $G$ be a noncylic abelian $p$-group. From the previously obtained results it follows that if $G$ is separable then $G$ is isomorphic to $C_p\\times C_{p^k}$ or $C_p\\times C_p\\times C_{p^k}$, where $p\\in \\{2,3\\}$ and $k\\geq 1$. We prove that the groups $D=C_p\\times C_{p^k}$ are separable whenever $p\\in \\{2,3\\}$. From this statement we deduce that a given Cayley graph over $D$ and a given Cayley graph over an arbitrary abelian group one can check whether these graphs are isomorphic in time $|D|^{O(1)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-12T16:21:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05C60",
      "20B35"
    ],
    "keywords": [
      "abelian p-groups",
      "schur ring",
      "cayley graph",
      "separability",
      "arbitrary abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}