{
  "id": "1103.1610",
  "version": "v3",
  "published": "2011-03-08T19:35:40.000Z",
  "updated": "2012-01-13T20:17:40.000Z",
  "title": "Cohomology and profinite topologies for solvable groups of finite rank",
  "authors": [
    "Karl Lorensen"
  ],
  "comment": "This paper supersedes arXiv:1009.2645v5: the two theorems in the introduction to the latter paper are both corollaries to Theorem 1.1 in the present paper. In the second version, Theorem 1.1 is expressed in a slightly more general form than in the first version",
  "journal": "Bull. Aust. Math. Soc. 86 (2012), 254-265",
  "categories": [
    "math.GR"
  ],
  "abstract": "Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\\infty}$. We show that if $G$ is nilpotent, then the pro-$p$ completion map $G\\to \\hat{G}_p$ induces an isomorphism $H^\\ast(\\hat{G}_p,M)\\to H^\\ast(G,M)$ for any discrete $\\hat{G}_p$-module $M$ of finite $p$-power order. For the general case, we prove that $G$ contains a normal subgroup $N$ of finite index such that the map $H^\\ast(\\hat{N}_p,M)\\to H^\\ast(N,M)$ is an isomorphism for any discrete $\\hat{N}_p$-module $M$ of finite $p$-power order. Moreover, if $G$ lacks any $C_{p^\\infty}$-sections, the subgroup $N$ enjoys some additional special properties with respect to its pro-$p$ topology.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-01-13T20:17:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F16",
      "20E18",
      "20F18"
    ],
    "keywords": [
      "solvable group",
      "profinite topologies",
      "finite rank",
      "power order",
      "cohomology"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.1610L"
    }
  }
}