{
  "id": "1607.02464",
  "version": "v1",
  "published": "2016-07-08T17:28:05.000Z",
  "updated": "2016-07-08T17:28:05.000Z",
  "title": "A Classification Theorem for Varieties Generated by Wreath Products of Groups",
  "authors": [
    "Vahagn H. Mikaelian"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "We suggest a criterion under which for a nilpotent group of finite exponent $A$ and for an abelian group $B$ the variety $var(A \\,Wr\\, B)$ generated by their wreath product $A \\,Wr\\, B$ is equal to the product of varieties $var(A)$ and $var(B)$ generated by $A$ and $B$. Namely the equality holds if and only if either the group $B$ is not of some non-zero exponent; or if $B$ is of a non-zero exponent $n$, and $B$ contains a subgroup isomorphic to $C_{d}^c \\times C_{n/d}^\\infty$, where $c$ is the nilpotency class of $A$, $d$ is the largest divisor of $n$ coprime with $m$, $C_{d}^c$ is the direct power of $c$ copies of the cycle $C_d$ of order $d$, $C_{n/d}^\\infty$ is the direct power of countably many copies of the cycle $C_{n/d}$ of order $n/d$. This criterion continues our previous work on cases when the similar criterions were given for wreath products of abelian groups or of finite groups. Also, this is a generalization of known results in literature, which solve the same problem for much more restricted cases. Some applications of the criterion are considered at the end of paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-08T17:28:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E22",
      "20E10",
      "20K01",
      "20K25",
      "20D15"
    ],
    "keywords": [
      "wreath product",
      "classification theorem",
      "abelian group",
      "non-zero exponent",
      "direct power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}