{
  "id": "0708.2819",
  "version": "v1",
  "published": "2007-08-21T11:59:53.000Z",
  "updated": "2007-08-21T11:59:53.000Z",
  "title": "On the cyclic subgroup separability of free products of two groups with amalgamated subgroup",
  "authors": [
    "E. V. Sokolov"
  ],
  "comment": "10 pages; for other papers of this author, see http://icu.ivanovo.ac.ru/tg-seminar",
  "journal": "Lobachevskii Journal of Mathematics. 11 (2002). 27-38",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a free product of two groups with amalgamated subgroup, $\\pi$ be either the set of all prime numbers or the one-element set \\{$p$\\} for some prime number $p$. Denote by $\\Sigma$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $\\pi$-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $\\pi$-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $\\pi^{\\prime}$-isolated, don't belong to $\\Sigma$. Some sufficient conditions are obtained for $\\Sigma$ to coincide with the family of all other $\\pi^{\\prime}$-isolated cyclic subgroups of group $G$. It is proved, in particular, that the residual $p$-finiteness of a free product with cyclic amalgamation implies the $p$-separability of all $p^{\\prime}$-isolated cyclic subgroups if the free factors are free or finitely generated residually $p$-finite nilpotent groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-08-21T11:59:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E06",
      "20E26"
    ],
    "keywords": [
      "free product",
      "cyclic subgroup separability",
      "amalgamated subgroup",
      "isolated cyclic subgroups",
      "prime number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.2819S"
    }
  }
}