{
  "id": "2203.00334",
  "version": "v1",
  "published": "2022-03-01T10:18:37.000Z",
  "updated": "2022-03-01T10:18:37.000Z",
  "title": "On closed subgroups of precompact groups",
  "authors": [
    "Salvador Hernández",
    "Dieter Remus",
    "F. Javier Trigos-Arrieta"
  ],
  "categories": [
    "math.GR",
    "math.GN"
  ],
  "abstract": "It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if $G$ is a subgroup of a compact Abelian group, and $S$ denotes those continuous homomorphisms from $G$ to the one-dimensional torus, then the topology on $G$ is the initial topology given by $S$. {Assume that $H$ is a subgroup of $G$. We study how} the choice of $S$ affects the topological placement and properties of $H$ in $G$. Among other results, we have {made significant} progress toward the solution of the following specific questions: How many totally bounded group topologies does $G$ admit such that $H$ is a closed (dense) subgroup? If $C_S$ denotes the poset of all subgroups of $G$ that are $S$-closed, ordered by inclusion, does $C_S$ has a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an \\textit{SC-group} (\\textit{topologically simple}, resp.) if all its subgroups are closed (if $G$ and $\\{e\\}$ are its only possible closed normal subgroups, resp.) {In addition, we investigate the following questions.} How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group $G$ admit?",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-01T10:18:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "precompact groups",
      "closed subgroups",
      "compact abelian group",
      "arbitrary abelian group",
      "closed normal subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}